A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

By Alex Wright

Abstract

We survey Mirzakhani’s work relating to Riemann surfaces, which spans about 20 papers. We target the discussion at a broad audience of nonexperts.

1. Introduction

This survey aims to be a tour through Maryam Mirzakhani’s remarkable work on Riemann surfaces, dynamics, and geometry. The star characters, all across mathematics and physics as well as in this survey, are the moduli spaces script upper M Subscript g comma n of Riemann surfaces.

Sections 2 through 10 all relate to Mirzakhani’s study of the size of these moduli spaces, as measured by the Weil–Petersson symplectic form (see the left side of Figure 1.1). Goldman has shown that many related moduli spaces also have a Weil–Petersson symplectic form, so this can be viewed as part of a broader story Reference Gol84. Even more important than the broader story, Mirzakhani’s study unlocks applications to the topology of script upper M Subscript g comma n , random surfaces of large genus, and even geodesics on individual hyperbolic surfaces.

Sections 11 to 19 reflect the philosophy that script upper M Subscript g comma n , despite being a totally inhomogeneous object, enjoys many of the dynamical properties of nicer spaces and even some of the dynamical miracles characteristic of homogeneous spaces (see the right side of Figure 1.1). The dynamics of group actions in turn clarify the geometry of script upper M Subscript g comma n and produce otherwise unattainable counting results.

Our goal is not to provide a comprehensive reference, but rather to highlight some of the most beautiful and easily understood ideas from the roughly 20 papers that constitute Mirzakhani’s work in this area. Very roughly speaking, we devote comparable time to each paper or closely related group of papers. This means in particular that we cannot proportionately discuss the longest paper Reference EM18, but on this topic the reader may see the surveys Reference Zor15Reference Wri16Reference Qui16. We include some open problems, and hope that we have succeeded in conveying the thriving legacy of Mirzakhani’s research.

We invite the reader to discover for themselves Mirzakhani’s five papers on combinatorics Reference MM95Reference Mir96Reference Mir98Reference MV15Reference MV17, where the author is not qualified to guide the tour.

We also omit comprehensive citations to work preceding Mirzakhani, suggesting instead that the reader may get off the tour bus at any time to find more details and context in the references and reboard later or to revisit the tour locations at a later date. Where possible, we give references to expository sources, which will be more useful to the learner than the originals. The reader who consults the references will be rewarded with views of the vast tapestry of important and beautiful work that Mirzakhani builds upon, something we can only offer tiny glimpses of here.

We hope that a second year graduate student who has previously encountered the definitions of hyperbolic space, Riemann surface, line bundle, symplectic manifold, etc., will be able to read and appreciate the survey, choosing not to be distracted by the occasional remark aimed at the experts.

Other surveys on Mirzakhani’s work include Reference Wol10Reference Wol13Reference Do13Reference McM14Reference Zor14Reference Zor15Reference Hua16Reference Wri16Reference Qui16Reference Mar17Reference Wri18. See also the issue of the Notices of the AMS that was devoted to Mirzakhani Reference Not18.

2. Preliminaries on Teichmüller theory

We begin with the beautiful and basic results that underlie most of Mirzakhani’s work.

2.1. Hyperbolic geometry and complex analysis

All surfaces are assumed to be orientable and connected. Any simply connected surface with a complete Riemannian metric of constant curvature negative 1 is isometric to the upper half-plane

double-struck upper H equals left-brace x plus i y bar y greater-than 0 right-brace

endowed with the hyperbolic metric

left-parenthesis d s right-parenthesis squared equals StartFraction left-parenthesis d x right-parenthesis squared plus left-parenthesis d y right-parenthesis squared Over y squared EndFraction period

Perhaps the most important miracle of low-dimensional geometry is that the group of orientation-preserving isometries of hyperbolic space is equal to the group of biholomorphisms of double-struck upper H . (Both are equal to the group upper P upper S upper L left-parenthesis 2 comma double-struck upper R right-parenthesis of Möbius transformations that stabilize the upper half-plane.)

Every oriented complete hyperbolic surface upper X has universal cover double-struck upper H , and the deck group acts on double-struck upper H via orientation-preserving isometries. Since these isometries are also biholomorphisms, this endows upper X with the structure of a Riemann surface, namely an atlas of charts to double-struck upper C whose transition functions are biholomorphisms.

Conversely, every Riemann surface upper X that is not simply connected, not double-struck upper C minus StartSet 0 EndSet , and not a torus has universal cover double-struck upper H , and the deck group acts on double-struck upper H via biholomorphisms. Since these biholomorphisms are also isometries, this endows upper X with a complete hyperbolic metric.

2.2. Cusps, geodesics, and collars

Suppose upper X is a complete hyperbolic surface. Each subset of upper X isometric to

StartSet z equals x plus i y element-of double-struck upper H colon y greater-than 1 EndSet slash mathematical left-angle z right-arrow from bar z plus 1 mathematical right-angle

is called a cusp. Each cusp has infinite diameter and finite volume. Distinct cusps are disjoint, and if upper X has finite area, then the complement of the cusps is compact.

Each cusp is biholomorphic to a punctured disc via the exponential map. If upper X has finite area, then upper X is biholomorphic to a compact Riemann surface minus a finite set of punctures, and the punctures are in bijection with the cusps.

Any closed curve on upper X not homotopic to a point or a loop around a cusp is isotopic to a unique closed geodesic. Unless otherwise stated, all closed curves we consider will be of this type. A closed geodesic is called simple if it does not intersect itself.

Gauss–Bonnet gives that any closed hyperbolic surface of genus g has area 2 pi left-parenthesis 2 g minus 2 right-parenthesis . This is the first indication that a hyperbolic surface cannot be “small”. Moreover, the Collar Lemma gives that any closed geodesic of length less than a universal constant is simple, and every short simple closed geodesic must be surrounded by a large embedded annulus known as its collar. As the length of the simple closed geodesic goes to zero, the size of its collar goes to infinity. See Figure 2.1.

2.3. Building a surface out of pants

A significant amount of this survey will concern hyperbolic surfaces with boundary. We will always assume that any surface with boundary that we consider can be isometrically embedded in a complete surface so that the boundary consists of a finite union of closed geodesics.

A hyperbolic sphere with three boundary components is known as a pair of pants, or simply as a pants. A fundamental fact gives that, for any three numbers upper L 1 comma upper L 2 comma upper L 3 greater-than 0 , there is a unique pants with these three boundary lengths. Each upper L Subscript i may also be allowed to be zero, in which case a pants, now somewhat degenerate, has a cusp instead of a boundary component.

One of the simplest ways to build a closed hyperbolic surface is by gluing together pants. For example, given two pants with the same boundary lengths, we may glue together the corresponding boundaries to obtain a closed genus 2 surface, as in Figure 2.2. In fact, the corresponding boundaries can be glued using different isometries from the circle to the circle, giving infinitely many genus 2 hyperbolic surfaces. More complicated surfaces can be obtained by gluing together more pants.

2.4. Teichmüller space and moduli space

We define moduli space script upper M Subscript g comma n formally as the set of equivalence classes of oriented genus g hyperbolic surfaces with n cusps labeled by StartSet 1 comma ellipsis comma n EndSet , where two surfaces are considered equivalent if they are isometric via an orientation-preserving isometry that respects the labels of the cusps. Equivalently, script upper M Subscript g comma n can be defined as the set of equivalence classes of genus  g Riemann surfaces with n punctures labeled by StartSet 1 comma ellipsis comma n EndSet , where two surfaces are considered equivalent if they are biholomorphic via a biholomorphism that respects the labels of the punctures.

We will follow the almost universal abuse of referring to a point in script upper M Subscript g comma n as a hyperbolic or Riemann surface, leaving out the notational bookkeeping of the equivalence class.

Teichmüller space script upper T Subscript g comma n is defined to be the set

script upper T Subscript g comma n Baseline equals left-brace left-parenthesis upper X comma left-bracket phi right-bracket right-parenthesis right-brace

of points upper X in script upper M Subscript g comma n , which as indicated we think of as hyperbolic or Riemann surfaces, equipped with a homotopy class left-bracket phi right-bracket of orientation-preserving homeomorphisms phi colon upper S Subscript g comma n Baseline right-arrow upper X from a fixed oriented topological surface upper S Subscript g comma n of genus g with n punctures. The homotopy class left-bracket phi right-bracket is called a marking, and one says that script upper T Subscript g comma n parametrizes marked hyperbolic or Riemann surfaces.

Let upper H o m e o Superscript plus Baseline left-parenthesis upper S Subscript g comma n Baseline right-parenthesis denote the group of orientation-preserving homeomorphisms of upper S Subscript g comma n that do not permute the punctures. This group acts on script upper T Subscript g comma n by precomposition with the marking. The subgroup upper H o m e o 0 Superscript plus Baseline left-parenthesis upper S Subscript g comma n Baseline right-parenthesis of homeomorphisms isotopic to the identity acts trivially, so the quotient

upper M upper C upper G Subscript g comma n Baseline equals upper H o m e o Superscript plus Baseline left-parenthesis upper S Subscript g comma n Baseline right-parenthesis slash upper H o m e o 0 Superscript plus Baseline left-parenthesis upper S Subscript g comma n Baseline right-parenthesis

acts on script upper T Subscript g comma n . This countable group is called the mapping class group, and

script upper M Subscript g comma n Baseline equals script upper T Subscript g comma n Baseline slash upper M upper C upper G Subscript g comma n Baseline period

Given upper L equals left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis element-of double-struck upper R Subscript plus Superscript n , we can similarly define script upper T Subscript g comma n Baseline left-parenthesis upper L right-parenthesis to be the Teichmüller space of oriented genus g hyperbolic surfaces with n boundary components of length upper L 1 comma ellipsis comma upper L Subscript n Baseline , and we define script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis to be the corresponding moduli space. Here upper S Subscript g comma n is replaced with a genus g surface with n boundary circles, and we define upper H o m e o Superscript plus Baseline left-parenthesis upper S Subscript g comma n Baseline right-parenthesis to be the orientation-preserving homeomorphisms that do not permute the boundary components.

It is sometimes convenient to allow upper L Subscript i Baseline equals 0 in the definitions above, in which case the corresponding boundary is replaced by a cusp. For example, using this convention script upper M Subscript g comma n Baseline equals script upper M Subscript g comma n Baseline left-parenthesis 0 comma ellipsis comma 0 right-parenthesis .

2.5. Classification of simple closed curves

Let alpha and beta be two different simple closed curves on upper S Subscript g that are nonseparating, in that cutting either curve does not disconnect the surface. In this case the result of cutting either alpha or beta is homeomorphic to a genus g minus 1 surface with two boundary curves, and hence they are homeomorphic to each other. This homeomorphism can be modified to give rise to a homeomorphism of upper S Subscript g that takes alpha to beta . In particular, we conclude that there is some f element-of upper M upper C upper G Subscript g such that f left-parenthesis left-bracket alpha right-bracket right-parenthesis equals left-bracket beta right-bracket , where left-bracket alpha right-bracket denotes the homotopy class of alpha .

Next suppose that alpha is a separating simple closed curve. In this case, upper S Subscript g Baseline minus alpha has two components, one of which is a surface of genus g 1 with one boundary component, and the other of which is a surface of genus g 2 equals g minus g 1 with one boundary component. If beta is another separating curve, then there is some f element-of upper M upper C upper G Subscript g such that f left-parenthesis left-bracket alpha right-bracket right-parenthesis equals left-bracket beta right-bracket if and only if the set StartSet g 1 comma g 2 EndSet arising from beta is the same as for alpha .

In summary, there is a single mapping class group orbit of nonseparating simple closed curves on upper S Subscript g , and there are left floor StartFraction g Over 2 EndFraction right floor mapping class group orbits of separating simple closed curves.

2.6. The twist flow

Let alpha be a simple closed curve on upper S Subscript g comma n that is not a loop around a cusp. We now introduce the twist flow upper T w Subscript t Superscript alpha on Teichmüller space as follows. It may be conceptually helpful to start by assuming t is small and positive.

For each point left-parenthesis upper X comma left-bracket phi right-bracket right-parenthesis element-of script upper T Subscript g comma n , we can consider the geodesic representative of phi left-parenthesis alpha right-parenthesis . Cut this geodesic to obtain a surface with two geodesic boundary components of equal length. Both of these components inherit an orientation from the surface (see Figure 2.3), and we will call the positive direction “left”.

Reglue the two components by the original identification, composed with a rotation by t , so that if two points p 1 comma p 2 on the two boundary components were originally identified, now p 1 is identified with the point t to the left of p 2 , and vice versa. See Figure 2.4.

If we use the notation upper T w Subscript t Superscript alpha Baseline left-parenthesis upper X comma left-bracket phi right-bracket right-parenthesis equals left-parenthesis upper X Subscript t Baseline comma left-bracket phi Subscript t Baseline right-bracket right-parenthesis , this regluing defines upper X Subscript t . The marking left-bracket phi Subscript t Baseline right-bracket is more subtle, and we will omit its definition. Here it will suffice to accept that, despite the fact that upper X Subscript t plus script l Sub Subscript alpha Subscript left-parenthesis upper X right-parenthesis Baseline equals upper X Subscript t , the twist path is injective, so left-parenthesis upper X Subscript t 1 Baseline comma left-bracket phi Subscript t 1 Baseline right-bracket right-parenthesis equals left-parenthesis upper X Subscript t 2 Baseline comma left-bracket phi Subscript t 2 Baseline right-bracket right-parenthesis if and only if t 1 equals t 2 .

2.7. Fenchel–Nielsen coordinates

Fix a pants decomposition of upper S Subscript g comma n . This is a collection of disjoint simple closed curves, such that cutting these curves gives a collection of topological pants. It turns out any such collection has 3 g minus 3 plus n curves, and we denote these curves left-brace alpha Subscript i Baseline right-brace Subscript i equals 1 Superscript 3 g minus 3 plus n . If n greater-than 0 , some of the pants will be degenerate, in that they will have a puncture instead of boundary circle. See Figure 2.5.

Given a marked hyperbolic surface left-parenthesis upper X comma phi right-parenthesis , we can consider the curves phi left-parenthesis alpha Subscript i Baseline right-parenthesis on upper X . Let script l Subscript alpha Sub Subscript i Baseline left-parenthesis upper X right-parenthesis denote the length of the geodesic homotopic to phi left-parenthesis alpha Subscript i Baseline right-parenthesis . For short, we write script l Subscript i to denote script l Subscript alpha Sub Subscript i Baseline left-parenthesis upper X right-parenthesis .

Each hyperbolic surface in script upper T Subscript g comma n can be obtained by gluing together the pants with the correct boundary lengths in the correct combinatorial pattern, but additional parameters are required in the construction to keep track of how the boundary curves are glued together.

Theorem 2.1 (Fenchel and Nielsen).

There are functions

tau Subscript i Baseline colon script upper T Subscript g comma n Baseline right-arrow double-struck upper R comma i equals 1 comma ellipsis comma 3 g minus 3 plus n comma

such that the map script upper T Subscript g comma n Baseline right-arrow double-struck upper R Subscript plus Superscript 3 g minus 3 plus n Baseline times double-struck upper R Superscript 3 g minus 3 plus n defined by

left-parenthesis script l 1 comma ellipsis comma script l Subscript 3 g minus 3 plus n Baseline comma tau 1 comma ellipsis comma tau Subscript 3 g minus 3 plus n Baseline right-parenthesis

is a homeomorphism, and so that for each i and all t ,

tau Subscript i Baseline left-parenthesis upper T w Subscript alpha Sub Subscript i Subscript Superscript t Baseline left-parenthesis upper X comma left-bracket phi right-bracket right-parenthesis right-parenthesis equals t plus tau Subscript i Baseline left-parenthesis upper X comma left-bracket phi right-bracket right-parenthesis comma

and all the other coordinates of upper T w Subscript alpha Sub Subscript i Superscript t Baseline left-parenthesis upper X comma left-bracket phi right-bracket right-parenthesis and left-parenthesis upper X comma left-bracket phi right-bracket right-parenthesis are the same.

The twist parameters tau Subscript i and the length parameters script l Subscript i are called Fenchel–Nielsen coordinates for Teichmüller space.

One can show that the mapping class group acts properly discontinuously on script upper T Subscript g comma n . In particular, the stabilizer of each point is finite. The quotient script upper M Subscript g comma n is thus an orbifold, which is similar to a manifold except that some points have neighborhoods homeomorphic to a neighborhood of the origin in double-struck upper R Superscript 6 g minus 6 plus 2 n quotiented by a finite group action.

Fenchel–Nielsen coordinates work similarly for script upper T Subscript g comma n Baseline left-parenthesis upper L right-parenthesis . Note there are no twist or length parameters for the geodesic boundary curves, since they have fixed lengths upper L Subscript i and are not glued to anything.

2.8. The Weil–Petersson symplectic structure

Fix a choice of Fenchel– Nielsen coordinates, and define

omega Subscript upper W upper P Baseline equals sigma-summation d script l Subscript i Baseline logical-and d tau Subscript i

to be the standard symplectic form in these coordinates on script upper T Subscript g comma n Baseline left-parenthesis upper L right-parenthesis .

Wolpert proved that this symplectic form is invariant under the action of the mapping class group. Hence it descends to a symplectic form omega Subscript upper W upper P on script upper M Subscript g comma n or script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis .⁠Footnote1 We define omega Subscript upper W upper P on moduli space so that its pullback to Teichmüller space is the standard symplectic form defined above. In other words, it is the standard symplectic form in local Fenchel–Nielsen coordinates on moduli space. This is sometimes called the “topologist’s definition”, and it ignores that script upper M Subscript g comma n may be considered as a stack, which is the algebro-geometric version of an orbifold. To reconcile with the algebro-geometric perspective without using stacks, one could also define the Weil–Petersson volume form on script upper M Subscript g comma n as the local pushforward of the Weil–Petersson volume form on Teichmüller space. The definitions give volume forms that are equal except for script upper M 2 and script upper M 1 left-parenthesis upper L right-parenthesis , where the reconciled volume form is half of the topologist’s. Every surface in those two moduli spaces has an involution symmetry. Independently of this issue, it is also common to include a separate factor of one half in the definition of omega Subscript upper W upper P for all g and n Reference Wol07, Section 5.

Wolpert also showed that in the case of script upper T Subscript g comma n , this symplectic form is twice the one arising from the Weil–Petersson Kähler structure on script upper T Subscript g comma n . This result is sometimes called Wolpert’s Magic Formula, since

the definition of the Weil–Petersson Kähler structure, although very natural, gives no hint of a relationship to Fenchel–Nielsen coordinates, and

it is surprising that the two-form sigma-summation d script l Subscript i logical-and d tau Subscript i , obtained from a pants decomposition, does not depend on which pants decomposition is used.

The associated Weil–Petersson volume form, which is the standard volume form in local Fenchel–Nielsen coordinates, is the most natural known notion of volume on moduli space. The Weil–Peterson volume of each moduli space is finite.

2.9. References

More details can be found in the books Reference FM12, Chapters 10, 12 and, for the Weil–Petersson symplectic structure, Reference Wol10, Chapter 3.

3. The volume of script upper M Subscript 1 comma 1

Mirzakhani discovered an elegant new computation of the volume of script upper M Subscript 1 comma 1 . We reproduce this computation, which is highlighted in the introduction to Reference Mir07b, since it was perhaps the first seed for her thesis. The starting point is the remarkable identity

sigma-summation Underscript alpha Endscripts StartFraction 1 Over 1 plus e Superscript script l Super Subscript alpha Superscript left-parenthesis upper X right-parenthesis Baseline EndFraction equals one half comma

of McShane Reference McS98, which gives that a certain sum involving the lengths script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis of all simple closed geodesics alpha on upper X element-of script upper M Subscript 1 comma 1 is independent of upper X element-of script upper M Subscript 1 comma 1 . In Section 5 we will explain where this identity comes from.

Let script upper M Subscript 1 comma 1 Superscript asterisk denote the infinite cover of script upper M Subscript 1 comma 1 parametrizing pairs left-parenthesis upper X comma alpha right-parenthesis , where upper X element-of script upper M Subscript 1 comma 1 and alpha is a simple closed geodesic on upper X . Mirzakhani’s computation begins

StartLayout 1st Row 1st Column one half upper V o l left-parenthesis script upper M Subscript 1 comma 1 Baseline right-parenthesis 2nd Column equals 3rd Column integral Underscript script upper M Subscript 1 comma 1 Endscripts sigma-summation Underscript alpha Endscripts StartFraction 1 Over 1 plus e Superscript script l Super Subscript upper X Superscript left-parenthesis alpha right-parenthesis Baseline EndFraction normal d normal upper V normal o normal l Subscript upper W upper P 2nd Row 1st Column Blank 2nd Column equals 3rd Column integral Underscript script upper M Subscript 1 comma 1 Superscript asterisk Baseline Endscripts StartFraction 1 Over 1 plus e Superscript script l Super Subscript upper X Superscript left-parenthesis alpha right-parenthesis Baseline EndFraction normal d normal upper V normal o normal l Subscript upper W upper P Baseline period EndLayout

This unfolding is justified because the fibers of the map script upper M Subscript 1 comma 1 Superscript asterisk Baseline right-arrow script upper M 1 are precisely the set of simple closed geodesics, which is the set being summed over in McShane’s identity.

Given a point left-parenthesis upper X comma alpha right-parenthesis element-of script upper M Subscript 1 comma 1 Superscript asterisk , we may cut upper X along alpha to get a hyperbolic sphere with two boundary components of length script l equals script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis , and one cusp. It is helpful to view this as a degenerate pants (see Figure 3.1), where one of the boundary curves has been replaced with a cusp.

For any script l greater-than 0 , there is a unique such pants, so the point left-parenthesis upper X comma alpha right-parenthesis element-of script upper M Subscript 1 comma 1 Superscript asterisk is uniquely determined by script l equals script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis and a twist parameter tau element-of left-bracket 0 comma script l right-parenthesis which controls how the two length script l boundary curves of the degenerate pants are glued together to give upper X .

Using this parametrization

script upper M Subscript 1 comma 1 Superscript asterisk Baseline asymptotically-equals left-brace left-parenthesis script l comma tau right-parenthesis colon script l greater-than 0 comma 0 less-than-or-equal-to tau less-than script l right-brace

and Wolpert’s formula omega Subscript upper W upper P Baseline equals d script l logical-and d tau , Mirzakhani concludes

StartLayout 1st Row 1st Column one half upper V o l script upper M Subscript 1 comma 1 2nd Column equals 3rd Column integral Subscript 0 Superscript normal infinity Baseline integral Subscript 0 Superscript script l Baseline StartFraction 1 Over 1 plus e Superscript script l Baseline EndFraction normal d tau normal d script l equals StartFraction pi squared Over 12 EndFraction period EndLayout

4. Integrating geometric functions over moduli space

In this section we give a key result from Reference Mir07b that gives a procedure for integrating certain functions over moduli space, generalizing the unfolding step in the previous section.

4.1. A special case

Let gamma be a simple nonseparating closed curve on a surface of genus g greater-than 2 . For a continuous function f colon double-struck upper R Subscript plus Baseline right-arrow double-struck upper R Subscript plus Baseline , we define a function f Subscript gamma Baseline colon script upper M Subscript g Baseline right-arrow double-struck upper R by

f Subscript gamma Baseline left-parenthesis upper X right-parenthesis equals sigma-summation Underscript left-bracket alpha right-bracket element-of upper M upper C upper G dot left-bracket gamma right-bracket Endscripts f left-parenthesis script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis right-parenthesis period

Here the sum is over the mapping class group orbit of the homotopy class left-bracket gamma right-bracket of gamma . Soon we will generalize this notation, but for the moment the subscript may seem strange: since there is only one mapping class group orbit of nonseparating simple closed curves, for the moment f Subscript gamma does not depend on gamma .

Recall that script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis is the moduli space of genus g minus 1 hyperbolic surfaces with two labeled boundary geodesics of length script l . We will give an outline of Mirzakhani’s proof that

integral Underscript script upper M Subscript g Baseline Endscripts f Subscript gamma Baseline left-parenthesis upper X right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P Baseline equals one half integral Subscript 0 Superscript normal infinity Baseline script l f left-parenthesis script l right-parenthesis upper V o l left-parenthesis script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis right-parenthesis normal d script l period

Define script upper M Subscript g Superscript gamma to be the set of pairs left-parenthesis upper X comma alpha right-parenthesis , where upper X element-of script upper M Subscript g and alpha is a geodesic with left-bracket alpha right-bracket element-of upper M upper C upper G dot left-bracket gamma right-bracket . The fibers of the map

script upper M Subscript g Superscript gamma Baseline right-arrow script upper M Subscript g Baseline comma left-parenthesis upper X comma alpha right-parenthesis right-arrow from bar upper X comma

correspond exactly to the set left-brace left-bracket alpha right-bracket element-of upper M upper C upper G dot left-bracket gamma right-bracket right-brace that is summed over in the definition of f Subscript gamma , and in fact

integral Underscript script upper M Subscript g Baseline Endscripts f Subscript gamma Baseline left-parenthesis upper X right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P Baseline equals integral Underscript script upper M Subscript g Superscript gamma Baseline Endscripts f left-parenthesis script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P Baseline period

Cutting upper X along alpha almost determines a point of script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis , except that the two boundary geodesics are not labeled. However, since there are two choices of labeling, we can say that there is a two-to-one map

left-brace left-parenthesis script l comma upper Y comma tau right-parenthesis colon script l greater-than 0 comma upper Y element-of script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis comma tau element-of double-struck upper R slash script l double-struck upper Z right-brace right-arrow script upper M Subscript g Superscript gamma Baseline comma

where the map glues together the two boundary components of upper Y with a twist determined by tau . Wolpert’s Magic Formula determines the pullback of the Weil–Petersson measure, and we get

StartLayout 1st Row 1st Column integral Underscript script upper M Subscript g Superscript gamma Endscripts f left-parenthesis script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P 2nd Column equals 3rd Column one half integral Subscript script l equals 0 Superscript normal infinity Baseline integral Subscript tau equals 0 Superscript script l Baseline integral Underscript script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis Endscripts f left-parenthesis script l right-parenthesis normal d times normal upper V normal o normal l Subscript upper W upper P Baseline normal d tau normal d script l 2nd Row 1st Column Blank 2nd Column equals 3rd Column one half integral Subscript 0 Superscript normal infinity Baseline script l f left-parenthesis script l right-parenthesis upper V o l left-parenthesis script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis right-parenthesis normal d script l period EndLayout

The case of g equals 2 is special, because every upper Y element-of script upper M Subscript 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis has an involution exchanging the two boundary components. Because of this involution, one cannot distinguish between the two choices of labeling the two boundary components, and the map that was two-to-one is now one-to-one. Thus, the same formula holds in genus 2 with the factor of one half removed.

4.2. The general case

A simple multi-curve, often just multi-curve for short, is a finite sum of disjoint simple closed curves with positive real weights, none of whose components are loops around a cusp. If gamma equals sigma-summation Underscript i equals 1 Overscript k Endscripts c Subscript i Baseline gamma Subscript i is a multi-curve, its length is defined by

script l Subscript gamma Baseline left-parenthesis upper X right-parenthesis equals sigma-summation Underscript i equals 1 Overscript k Endscripts c Subscript i Baseline script l Subscript gamma Sub Subscript i Subscript Baseline left-parenthesis upper X right-parenthesis period

We define f Subscript gamma for multi-curves in the same way as above and note that f Subscript gamma in fact only depends on the mapping class group orbit of left-bracket gamma right-bracket .

Suppose that cutting the geodesic representative of gamma decomposes upper X element-of script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis into s connected components upper X 1 comma ellipsis comma upper X Subscript s Baseline , and that

upper X Subscript j has genus g Subscript j ,

upper X Subscript j has n Subscript j boundary components, and

the lengths of the boundary components of upper X Subscript j are given by normal upper Lamda Subscript j Baseline element-of double-struck upper R Subscript plus Superscript n Super Subscript j .

If we set script l Subscript i Baseline equals script l Subscript gamma Sub Subscript i Baseline left-parenthesis upper X right-parenthesis , then all the entries of each normal upper Lamda Subscript j are from StartSet upper L 1 comma ellipsis comma upper L Subscript n Baseline EndSet (if they correspond to the original boundary of upper X ) or StartSet script l 1 comma ellipsis comma script l Subscript k Baseline EndSet (if they correspond to the new boundary created by cutting gamma ).

Theorem 4.1 (Mirzakhani’s Integration Formula).

For any multi-curve gamma equals sigma-summation Underscript i equals 1 Overscript k Endscripts c Subscript i Baseline gamma Subscript i ,

StartLayout 1st Row 1st Column Blank 2nd Column integral Underscript script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis Endscripts f Subscript gamma Baseline normal d normal upper V normal o normal l Subscript upper W upper P 2nd Row 1st Column Blank 2nd Column equals iota Subscript gamma Baseline integral Underscript script l equals left-parenthesis script l 1 comma ellipsis comma script l Subscript k Baseline right-parenthesis element-of double-struck upper R Subscript plus Superscript k Baseline Endscripts script l 1 midline-horizontal-ellipsis script l Subscript k Baseline f left-parenthesis c 1 script l 1 plus midline-horizontal-ellipsis plus c Subscript k Baseline script l Subscript k Baseline right-parenthesis product Underscript j equals 1 Overscript s Endscripts upper V o l left-parenthesis script upper M Subscript g Sub Subscript j Subscript comma n Sub Subscript j Subscript Baseline left-parenthesis normal upper Lamda Subscript j Baseline right-parenthesis right-parenthesis normal d script l comma EndLayout

where iota Subscript gamma Baseline element-of double-struck upper Q Subscript plus is an explicit constant.⁠Footnote2 Slightly different values of the constant have been recorded in different places in the literature. We believe the correct constant is iota Subscript gamma Baseline equals StartFraction 1 Over 2 Superscript upper M Baseline left-bracket upper S t a b left-parenthesis gamma right-parenthesis colon mathematical left-angle upper S comma intersection Underscript i equals 1 Overscript k Endscripts upper S t a b Superscript plus Baseline left-parenthesis gamma Subscript i Baseline right-parenthesis mathematical right-angle right-bracket EndFraction comma where upper M is the number of i such that gamma Subscript i bounds a torus with no other boundary components and not containing any other component of gamma , upper S t a b left-parenthesis gamma right-parenthesis is the stabilizer of the weighted multi-curve gamma , upper S t a b Superscript plus Baseline left-parenthesis gamma Subscript i Baseline right-parenthesis is the subgroup of the mapping class group that fixes gamma Subscript i and its orientation, and upper S is the kernel of the action of the mapping class group on Teichmüller space. (Note that upper S is trivial except in the case when left-parenthesis g comma n right-parenthesis is left-parenthesis 1 comma 1 right-parenthesis or left-parenthesis 2 comma 0 right-parenthesis , in which case it has size 2 and is central.) Given two subgroups upper H 1 comma upper H 2 , we write mathematical left-angle upper H 1 comma upper H 2 mathematical right-angle for the subgroup they generate. The number iota Subscript gamma Superscript negative 1 arises as the degree of a measurable map from script upper P equals left-brace left-parenthesis script l comma upper Y comma tau right-parenthesis colon script l element-of double-struck upper R Subscript plus Superscript k Baseline comma upper Y element-of product Underscript j equals 1 Overscript s Endscripts script upper M Subscript g Sub Subscript j Subscript comma n Sub Subscript j Subscript Baseline left-parenthesis normal upper Lamda Subscript j Baseline right-parenthesis comma tau element-of product Underscript i equals 1 Overscript k Endscripts double-struck upper R slash script l Subscript i Baseline double-struck upper Z right-brace to the space script upper M Subscript g comma n Superscript gamma Baseline left-parenthesis upper L right-parenthesis of pairs left-parenthesis upper X comma alpha right-parenthesis , where upper X element-of script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis and alpha is a multi-geodesic with left-bracket alpha right-bracket element-of upper M upper C upper G dot left-bracket gamma right-bracket . This natural map factors through the space script upper M Subscript g comma n Superscript gamma comma plus Baseline left-parenthesis upper L right-parenthesis of (ordered) tuples left-parenthesis upper X comma alpha 1 comma ellipsis comma alpha Subscript k Baseline right-parenthesis , where upper X element-of script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis , and the alpha Subscript i are disjoint oriented geodesics with left-bracket sigma-summation c Subscript i Baseline alpha Subscript i Baseline right-bracket element-of upper M upper C upper G dot left-bracket gamma right-bracket . The degree of script upper P right-arrow script upper M Subscript g comma n Superscript gamma comma plus Baseline left-parenthesis upper L right-parenthesis is 2 Superscript upper M and the degree of script upper M Subscript g comma n Superscript gamma comma plus Baseline left-parenthesis upper L right-parenthesis right-arrow script upper M Subscript g comma n Superscript gamma Baseline left-parenthesis upper L right-parenthesis is the remaining index factor.

5. Generalizing McShane’s identity

The starting point for Mirzakhani’s volume computations is the following result proven in Reference Mir07b. It relies on two explicit functions script upper D comma script upper R colon double-struck upper R Subscript plus Superscript 3 Baseline right-arrow double-struck upper R Subscript plus Baseline whose exact definitions are omitted here.

Theorem 5.1.

For any hyperbolic surface upper X with n geodesic boundary circles beta 1 comma ellipsis comma beta Subscript n Baseline of lengths upper L 1 comma ellipsis comma upper L Subscript n Baseline ,

sigma-summation Underscript gamma 1 comma gamma 2 Endscripts script upper D left-parenthesis upper L 1 comma script l Subscript upper X Baseline left-parenthesis gamma 1 right-parenthesis comma script l Subscript upper X Baseline left-parenthesis gamma 2 right-parenthesis right-parenthesis plus sigma-summation Underscript i equals 2 Overscript n Endscripts sigma-summation Underscript gamma Endscripts script upper R left-parenthesis upper L 1 comma upper L Subscript i Baseline comma script l Subscript upper X Baseline left-parenthesis gamma right-parenthesis right-parenthesis equals upper L 1 comma

where the first sum is over all pairs of closed geodesics gamma 1 comma gamma 2 bounding a pants with beta 1 , and the second sum is over all simple closed geodesics gamma bounding a pants with beta 1 and beta Subscript i (see Figure 5.1 right-parenthesis period

By studying the asymptotics of this formula when some upper L Subscript i Baseline right-arrow 0 , it is possible to derive a related formula in the case when the boundary beta Subscript i is replaced with a cusp. In the case when all beta Subscript i are replaced with cusps, Mirzakhani recovers identities due to McShane Reference McS98, including the identity given in Section 3. Thus, Mirzakhani refers to Theorem 5.1 as the generalized McShane identities.

Idea of the proof of Theorem 5.1.

Let upper F be the set of points x on beta 1 from which the unique geodesic ray gamma Subscript x beginning at x and perpendicular to the boundary continues forever without intersecting itself or hitting the boundary. By a result of Birman and Series, upper F has measure 0, reflecting the fact that most geodesic rays intersect themselves Reference BS85.

It is easy to see that beta 1 minus upper F is open and, hence, is a countable union of disjoint intervals left-parenthesis a Subscript h Baseline comma b Subscript h Baseline right-parenthesis .

Mirzakhani shows that the geodesics gamma Subscript a Sub Subscript h and gamma Subscript b Sub Subscript h both spiral toward either a simple closed curve or a boundary component other than beta 1 , as in Figure 5.2. There is a unique pants upper P with geodesic boundary containing gamma Subscript a Sub Subscript h and gamma Subscript b Sub Subscript h .

Each pants upper P is associated with one or more intervals left-parenthesis a Subscript h Baseline comma b Subscript h Baseline right-parenthesis , and the sum of the lengths of these intervals characterizes the functions script upper D and script upper R . Having computed these functions, the identity is equivalent to sigma-summation Underscript h Endscripts StartAbsoluteValue b Subscript h Baseline minus a Subscript h Baseline EndAbsoluteValue equals upper L 1 .

5.1. References

See Reference BT16 for a survey of related identities that have been proven since Mirzakhani’s work. Of special note is that there is a related identity for closed surfaces Reference LT14.

6. Computation of volumes using McShane identities

We now outline how Mirzakhani used her integration formula and the generalized McShane identities to recursively compute the Weil–Petersson volumes of script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis Reference Mir07b. Except in the case of upper L equals left-parenthesis 0 comma ellipsis comma 0 right-parenthesis , script upper M Subscript 0 comma 4 Baseline left-parenthesis upper L right-parenthesis and script upper M 1 left-parenthesis upper L right-parenthesis , these volumes were unknown before Mirzakhani’s work.

As in Section 3, we begin by integrating the generalized McShane identity to obtain

StartLayout 1st Row 1st Column upper L 1 upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis right-parenthesis equals 2nd Column integral Underscript script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis Endscripts sigma-summation Underscript gamma 1 comma gamma 2 Endscripts script upper D left-parenthesis upper L 1 comma script l Subscript upper X Baseline left-parenthesis gamma 1 right-parenthesis comma script l Subscript upper X Baseline left-parenthesis gamma 2 right-parenthesis right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P 2nd Row 1st Column Blank 2nd Column plus integral Underscript script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis Endscripts sigma-summation Underscript i equals 2 Overscript n Endscripts sigma-summation Underscript gamma Endscripts script upper R left-parenthesis upper L 1 comma upper L Subscript i Baseline comma script l Subscript upper X Baseline left-parenthesis gamma right-parenthesis right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P Baseline period EndLayout

In fact, StartFraction partial-differential Over partial-differential x EndFraction script upper D left-parenthesis x comma y comma z right-parenthesis and StartFraction partial-differential Over partial-differential x EndFraction script upper R left-parenthesis x comma y comma z right-parenthesis are nicer functions than script upper D and script upper R , so Mirzakhani considers the StartFraction partial-differential Over partial-differential upper L 1 EndFraction derivative of this identity.

Let us consider just the sum

sigma-summation Underscript gamma Endscripts StartFraction partial-differential Over partial-differential upper L 1 EndFraction script upper R left-parenthesis upper L 1 comma upper L 2 comma script l Subscript upper X Baseline left-parenthesis gamma right-parenthesis right-parenthesis

over all simple closed geodesics gamma which bound a pants with beta 1 and beta 2 . The set of such gamma is one mapping class group orbit, so we may apply Mirzakhani’s Integration Formula to get

StartLayout 1st Row 1st Column Blank 2nd Column integral Underscript script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis Endscripts sigma-summation Underscript gamma Endscripts StartFraction partial-differential Over partial-differential upper L 1 EndFraction script upper R left-parenthesis upper L 1 comma upper L Subscript i Baseline comma script l Subscript upper X Baseline left-parenthesis gamma right-parenthesis right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P 2nd Row 1st Column Blank 2nd Column equals iota Subscript gamma Baseline integral Underscript double-struck upper R Subscript plus Baseline Endscripts script l StartFraction partial-differential Over partial-differential upper L 1 EndFraction script upper R left-parenthesis upper L 1 comma upper L Subscript i Baseline comma script l right-parenthesis upper V o l left-parenthesis script upper M Subscript g comma n minus 1 Baseline left-parenthesis script l comma upper L 2 comma ellipsis comma upper L Subscript n Baseline right-parenthesis right-parenthesis normal d script l period EndLayout

Note that the surfaces in script upper M Subscript g comma n minus 1 Baseline left-parenthesis script l comma upper L 2 comma ellipsis comma upper L Subscript n Baseline right-parenthesis are smaller than those in script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis in that they have one less pants in a pants decomposition.

The sum over pairs gamma 1 comma gamma 2 is similar, but more complicated because the set of multi-curves gamma 1 plus gamma 2 that arise consists of a finite but possibly large number of mapping class group orbits.

This produces an expression for StartFraction partial-differential Over partial-differential upper L 1 EndFraction upper L 1 upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis right-parenthesis as a finite sum of integrals involving volumes of smaller moduli spaces. Mirzakhani was able to compute these integrals, allowing her to compute upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis right-parenthesis recursively. These computations imply in particular that upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis right-parenthesis is a polynomial in the upper L Subscript i Superscript 2 whose coefficients are positive rational multiples of powers of pi , which we will reprove from a different point of view in the next section.

6.1. References

Mirzakhani’s recursions are concisely presented in terms of the coefficients of the polynomials in Reference Mir13, Section 3.1 and Reference MZ15, Section 2.⁠Footnote3 Due to a different convention, in these papers the volumes of script upper M 2 and script upper M Subscript 1 comma 1 Baseline left-parenthesis upper L right-parenthesis are half what our conventions give. In the second line of Reference Mir13, Section 3.1, there is a typo that should be corrected as d 0 equals 3 g minus 3 plus n minus StartAbsoluteValue d EndAbsoluteValue . In Reference MZ15, Equation 2.13, there is a typo that should be corrected as StartAbsoluteValue upper I square-cup upper J EndAbsoluteValue equals StartSet 2 comma ellipsis comma n EndSet . Using these recursions to compute the volume polynomials is rather slow, because of the combinatorial explosion in high genus of the number of different moduli spaces that arise from recursively cutting along geodesics. Zograf has given a faster algorithm Reference Zog08.

Mirzakhani’s results do not directly allow for the computation of upper V o l left-parenthesis script upper M Subscript g Baseline right-parenthesis . However these volumes were previously known via intersection theory. They can also be recovered via the remarkable formula

2 pi i left-parenthesis 2 g minus 2 right-parenthesis upper V o l left-parenthesis script upper M Subscript g Baseline right-parenthesis equals StartFraction partial-differential upper V o l left-parenthesis partial-differential script upper M Subscript g comma 1 Baseline right-parenthesis Over partial-differential upper L EndFraction left-parenthesis 2 pi i right-parenthesis

proven in Reference DN09.

It would be interesting to recompute upper V o l left-parenthesis script upper M Subscript g Baseline right-parenthesis using Mirzakhani’s strategy and the identity for closed surfaces in Reference BT16.

Mirzakhani’s recursions fit into the framework of topological recursions Reference Eyn14.

7. Computation of volumes using symplectic reduction

We now give Mirzakhani’s second point of view on Weil–Petersson volumes, from Reference Mir07c.

7.1. A larger moduli space

Consider the moduli space ModifyingAbove script upper M With caret Subscript g comma n of genus g Riemann surfaces with n geodesic boundary circles with a marked point on each boundary circle. This moduli space has dimension 2 n greater than that of script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis , because the length of each boundary circle can vary, and the marked point on each boundary circle can vary.

ModifyingAbove script upper M With caret Subscript g comma n admits a version of local Fenchel–Nielsen coordinates, where in addition to the usual Fenchel–Nielsen coordinates there is a length parameter script l Subscript i for each boundary circle and a parameter that keeps track of the position the each marked point on each boundary circle. The parameters keeping track of the marked points are thought of as twist parameters. The space ModifyingAbove script upper M With caret Subscript g comma n also has a Weil–Peterson form ModifyingAbove omega With caret Subscript upper W upper P , which is still described by Wolpert’s Magic Formula, meaning that it is the standard symplectic form in any system of local Fenchel–Nielsen coordinates.

Consider now the function mu colon ModifyingAbove script upper M With caret Subscript g comma n Baseline right-arrow double-struck upper R Subscript plus Superscript n defined by

mu equals left-parenthesis StartFraction script l 1 squared Over 2 EndFraction comma ellipsis comma StartFraction script l Subscript n Superscript 2 Baseline Over 2 EndFraction right-parenthesis period

The reason for this definition will become clear in the next subsection.

Let upper S Superscript 1 Baseline equals double-struck upper R slash double-struck upper Z , and consider the left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n action that moves the marked points along the boundary circles. Each level set mu Superscript negative 1 Baseline left-parenthesis upper L 1 squared slash 2 comma ellipsis comma upper L Subscript n Superscript 2 Baseline slash 2 right-parenthesis is invariant under the left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n action, and the quotient is the space script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis with fixed boundary lengths and no marked points on the boundary. That is,

script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis equals mu Superscript negative 1 Baseline left-parenthesis upper L 1 squared slash 2 comma ellipsis comma upper L Subscript n Superscript 2 Baseline slash 2 right-parenthesis slash left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n Baseline period

7.2. Symplectic reduction

We now review a version of the Duistermaat–Heckman Theorem in symplectic geometry, as it applies to ModifyingAbove script upper M With caret Subscript g comma n .

The symplectic form ModifyingAbove omega With caret Subscript upper W upper P on ModifyingAbove script upper M With caret Subscript g comma n provides a nondegenerate bilinear form on each tangent space to ModifyingAbove script upper M With caret Subscript g comma n . This gives an identification between the tangent space and its dual, and hence between vector fields and one-forms.

Any function upper H on ModifyingAbove script upper M With caret Subscript g comma n determines a one-form d upper H and hence also a vector field upper V Subscript upper H defined via this duality. This duality is recorded symbolically as

d upper H equals ModifyingAbove omega With caret Subscript upper W upper P Baseline left-parenthesis upper V Subscript upper H Baseline comma dot right-parenthesis period

The flow in the vector field upper V Subscript upper H is called the Hamiltonian flow of upper H , and upper H is called the Hamiltonian function.

To begin, take n equals 1 . Let script l 1 colon ModifyingAbove script upper M With caret Subscript g comma 1 Baseline right-arrow double-struck upper R Subscript plus Baseline denote the length of the unique boundary circle, and let tau 1 denote the twist coordinate giving the position of the marked point. The upper S Superscript 1 action on ModifyingAbove script upper M With caret Subscript g comma n discussed above is simply given by tau 1 right-arrow from bar tau 1 plus t script l 1 , where t element-of double-struck upper R slash double-struck upper Z , and hence is generated by the vector field script l 1 partial-differential Subscript tau 1 . Wolpert’s Magic Formula gives

ModifyingAbove omega With caret Subscript upper W upper P Baseline left-parenthesis script l 1 partial-differential Subscript tau 1 Baseline comma dot right-parenthesis equals script l 1 d script l 1 period

If upper H equals script l 1 squared slash 2 , then d upper H equals script l 1 d script l 1 , so the upper S Superscript 1 action is Hamiltonian with Hamiltonian function upper H .

Now take n greater-than 1 . Then the i th coordinate upper S Superscript 1 action on ModifyingAbove script upper M With caret Subscript g comma n , which moves the position of the marked point on the i th boundary circle, is Hamiltonian with Hamiltonian function script l Subscript i Superscript 2 Baseline slash 2 . Using a natural way to combine different Hamiltonian functions into a single function called the moment map, one says that the left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n action on ModifyingAbove script upper M With caret Subscript g comma n is Hamiltonian with moment map mu given above.

For every xi equals left-parenthesis upper L 1 squared slash 2 comma ellipsis comma upper L Subscript n Superscript 2 Baseline slash 2 right-parenthesis , the space mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis is the manifold parametrizing surfaces in script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis with a marked point on each boundary circle, and as we have discussed

mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis slash left-parenthesis upper S 1 right-parenthesis Superscript n Baseline equals script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis period

The fact that this quotient is a symplectic manifold is an instance of a general phenomenon called symplectic reduction.

Return to the case n equals 1 . Then mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis is a principal circle bundle over script upper M Subscript g comma 1 Baseline left-parenthesis upper L 1 right-parenthesis . (A principal upper S Superscript 1 bundle is a bundle with an action of upper S Superscript 1 that is simply transitive on fibers.) Let us call this circle bundle script upper C 1 .

Generalizing this to n greater-than 1 , we see that mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis right-arrow script upper M Subscript g comma 1 Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis is a product of n circle bundles script upper C Subscript i Baseline comma i equals 1 comma ellipsis comma n . Here script upper C Subscript i can be defined as the spaces of surfaces in script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis together with just a single marked point on the i th boundary circle (and no marked points on any of the other boundary circles).

Theorem 7.1 (Duistermaat–Heckman Theorem).

For any fixed xi and for t equals left-parenthesis t 1 comma ellipsis comma t Subscript n Baseline right-parenthesis element-of double-struck upper R Superscript n small enough, there exists a diffeomorphism

phi Subscript t Baseline colon mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis slash left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n Baseline right-arrow mu Superscript negative 1 Baseline left-parenthesis xi plus t right-parenthesis slash left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n

such that

phi Subscript t Superscript asterisk Baseline left-parenthesis omega Subscript upper W upper P Baseline right-parenthesis equals omega Subscript upper W upper P Baseline plus sigma-summation Underscript i equals 1 Overscript n Endscripts t Subscript i Baseline c 1 left-parenthesis script upper C Subscript i Baseline right-parenthesis comma

where c 1 left-parenthesis script upper C Subscript i Baseline right-parenthesis is the first Chern class of the circle bundle script upper C Subscript i over mu Superscript negative 1 Baseline left-parenthesis xi plus t right-parenthesis slash left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n . Here, on the left-hand side omega Subscript upper W upper P refers to the Weil–Petersson form on mu Superscript negative 1 Baseline left-parenthesis xi plus t right-parenthesis slash left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n , and on the right-hand side it refers to the Weil–Petersson form on mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis slash left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n .

The reader unfamiliar with Chern classes may in fact take this theorem to be the definition for this survey; we will not use any other properties of Chern classes.

Part of Theorem 7.1 is powered by a relative of the Darboux Theorem. The Darboux Theorem states that a neighborhood of any point in a symplectic manifold is symplectomorphic to the simplest thing you could guess it to be, namely a neighbourhood in a vector space with the standard symplectic form.

Here, a neighborhood of mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis is topologically mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis times left-parenthesis negative delta comma delta right-parenthesis Superscript n , and one can create a guess for what the symplectic form ModifyingAbove omega With caret Subscript upper W upper P might look like on mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis times left-parenthesis negative delta comma delta right-parenthesis Superscript n , using omega Subscript upper W upper P on mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis slash left-parenthesis upper S Superscript 1 Baseline right-parenthesis Superscript n and curvature forms for the circle bundles. The Equivariant Coisotropic Reduction Theorem, which is the relative of the Darboux Theorem we referred to, says that this guess is in fact symplectomorphic to a neighborhood of mu Superscript negative 1 Baseline left-parenthesis xi right-parenthesis in ModifyingAbove script upper M With caret Subscript g comma n .

7.3. Computations of volumes

From Theorem 7.1, upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis right-parenthesis is a polynomial in a small neighborhood of any left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis , and hence is globally a polynomial. By considering xi equals left-parenthesis epsilon squared slash 2 comma ellipsis comma epsilon squared slash 2 right-parenthesis , we get

StartLayout 1st Row 1st Column Blank 2nd Column upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis right-parenthesis 2nd Row 1st Column Blank 2nd Column equals StartFraction 1 Over left-parenthesis 3 g minus 3 plus n right-parenthesis factorial EndFraction integral Underscript script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis Endscripts omega Subscript upper W upper P Superscript 3 g minus 3 plus n Baseline 3rd Row 1st Column Blank 2nd Column equals StartFraction 1 Over left-parenthesis 3 g minus 3 plus n right-parenthesis factorial EndFraction integral Underscript script upper M Subscript g comma n Baseline left-parenthesis epsilon comma ellipsis comma epsilon right-parenthesis Endscripts left-parenthesis omega Subscript upper W upper P Baseline plus sigma-summation StartFraction upper L Subscript i Superscript 2 Baseline minus epsilon squared Over 2 EndFraction c 1 left-parenthesis script upper C Subscript i Baseline right-parenthesis right-parenthesis Superscript 3 g minus 3 plus n Baseline period EndLayout

Note that Theorem 7.1 only directly gives this for upper L Subscript i close to epsilon , but since the volume is a polynomial, it must in fact be true for all upper L Subscript i . Note also that omega Subscript upper W upper P denotes the Weil–Petersson symplectic form on script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis in the first integral and on script upper M Subscript g comma n Baseline left-parenthesis epsilon comma ellipsis comma epsilon right-parenthesis in the second integral.

Taking a limit as epsilon right-arrow 0 , Mirzakhani obtains

upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis right-parenthesis equals StartFraction 1 Over left-parenthesis 3 g minus 3 plus n right-parenthesis factorial EndFraction integral Underscript script upper M Subscript g comma n Baseline Endscripts left-parenthesis omega Subscript upper W upper P Baseline plus sigma-summation StartFraction upper L Subscript i Superscript 2 Baseline Over 2 EndFraction c 1 left-parenthesis script upper C Subscript i Baseline right-parenthesis right-parenthesis Superscript 3 g minus 3 plus n Baseline period

Here script upper C Subscript i can be defined as the circle of points on a horocycle of size 1 about the i th cusp.

One can first interpret this integral in terms of the differential forms representing c 1 left-parenthesis script upper C Subscript i Baseline right-parenthesis produced by the proof of Theorem 7.1. These differential forms, and the circle bundles script upper C Subscript i , extend continuously to a natural compactification script upper M overbar Subscript g comma n constructed by Deligne and Mumford, and so one can and typically does replace script upper M Subscript g comma n with script upper M overbar Subscript g comma n as the space to be integrated over. This allows a more topological interpretation of the integral as the pairing of a class in upper H Superscript 6 g minus 6 plus 2 n Baseline left-parenthesis script upper M overbar Subscript g Baseline right-parenthesis with the fundamental class of script upper M overbar Subscript g comma n .

In summary, we have the following.

Theorem 7.2.

The volume of script upper M Subscript g comma n Baseline left-parenthesis upper L 1 comma ellipsis comma upper L Subscript n Baseline right-parenthesis is a polynomial

sigma-summation Underscript StartAbsoluteValue alpha EndAbsoluteValue less-than-or-equal-to 3 g minus 3 plus n Endscripts upper C Subscript g Baseline left-parenthesis alpha right-parenthesis upper L Superscript 2 alpha

whose coefficients upper C Subscript g Baseline left-parenthesis alpha right-parenthesis are rational multiples of integrals of powers of the Chern classes c 1 left-parenthesis script upper C Subscript i Baseline right-parenthesis and the Weil–Petersson symplectic form. Here alpha equals left-parenthesis alpha 1 comma ellipsis comma alpha Subscript n Baseline right-parenthesis , StartAbsoluteValue alpha EndAbsoluteValue equals sigma-summation alpha Subscript i , and upper L Superscript 2 alpha Baseline equals product upper L Subscript i Superscript 2 alpha Super Subscript i .

We will discuss a number of interesting results and open problems about these polynomials in Section 10.

7.4. References

For more on the material relating to symplectic reduction, see, for example, Reference CdS01, Chapters 22, 23, 30.2.

The work of Mirzakhani suggests some similarities between moduli spaces of Riemann surfaces and spaces of representations of surface groups into compact Lie groups modulo conjugacy. These spaces of representations are also known as character varieties or moduli spaces of stable bundles. Mirzakhani points out the connection between her techniques and those used previously by Witten and others in this context Reference Wit92. See the citations in Mirzakhani’s papers and the survey Reference Jef05 for more details.

8. Witten’s conjecture

We now describe Mirzakhani’s proof of Witten’s conjecture Reference Mir07c. This brings us to the algebro-geometric perspective on the coefficients upper C Subscript g Baseline left-parenthesis alpha right-parenthesis from Theorem 7.2.

8.1. Intersection theory

Let script upper C Subscript i Baseline overbar and omega overbar Subscript upper W upper P denote the extensions of script upper C Subscript i and omega from script upper M Subscript g comma n to script upper M overbar Subscript g comma n .

The class c 1 left-parenthesis script upper C Subscript i Baseline overbar right-parenthesis element-of upper H squared left-parenthesis script upper M overbar Subscript g comma n Baseline comma double-struck upper Q right-parenthesis is typically denoted psi Subscript i , and it is much studied. One often defines psi Subscript i as the first Chern class of a line bundle script upper L Subscript i called the relative cotangent bundle at the i th marked point.

Wolpert showed that the cohomology class left-bracket omega overbar Subscript upper W upper P Baseline right-bracket element-of upper H squared left-parenthesis script upper M overbar Subscript g comma n Baseline right-parenthesis is equal to 2 pi squared kappa 1 , where kappa 1 element-of upper H squared left-parenthesis script upper M overbar Subscript g comma n Baseline comma double-struck upper Q right-parenthesis is the much-studied first kappa class Reference Wol83. As a result, all of the coefficients upper C Subscript g Baseline left-parenthesis alpha right-parenthesis from Theorem 7.2 are in double-struck upper Q left-bracket pi squared right-bracket .

The compactification script upper M overbar Subscript g comma n is an algebraic variety and a smooth orbifold, and the classes psi Subscript i and kappa 1 can be thought of as dual to (equivalence classes of) divisors, which are linear combinations of subvarieties of complex codimension 1. The intersection of two such classes, if transverse, has complex codimension 2, and similarly the intersection dimension Subscript double-struck upper C Baseline script upper M overbar Subscript g comma n Baseline equals 3 g minus 3 plus n of them, if transverse, is a finite collection of points. Integrals of a product of 3 g minus 3 plus n of the psi Subscript i and kappa 1 classes, which up to factors are exactly the coefficients upper C Subscript g Baseline left-parenthesis alpha right-parenthesis , count the number of points of intersection. Thus, they are called intersection numbers. See the book Reference LZ04, Chapter 4.6 for some example computations using this point of view.

It is hard for the uninitiated to fathom how much useful information such intersection numbers can contain, so we pause to give just a few points of motivation.

They are central to the study of the geometry of script upper M overbar Subscript g comma n .

By Theorem 7.1 they determine Weil–Petersson volumes. Later we will see that these volumes can be used to understand the geometry of Weil–Petersson random surfaces.

They appear in theoretical physics Reference Wit91.

They determine counts of combinatorial objects called ribbon graphs Reference Kon92.

They determine Hurwitz numbers, which count certain branched coverings of the sphere or, equivalently, factorizations of permutations into transpositions Reference ELSV01.

8.2. A generating function for intersection numbers

Make the notational convention

mathematical left-angle tau Subscript d 1 Baseline midline-horizontal-ellipsis tau Subscript d Sub Subscript n Subscript Baseline mathematical right-angle Subscript g Baseline equals integral Underscript script upper M overbar Subscript g comma n Baseline Endscripts psi 1 Superscript d 1 Baseline midline-horizontal-ellipsis psi Subscript n Superscript d Super Subscript n Superscript Baseline period

Unless sigma-summation d Subscript i Baseline equals 3 g minus 3 plus n , this is defined to be zero. Note that mathematical left-angle tau Subscript d 1 Baseline midline-horizontal-ellipsis tau Subscript d Sub Subscript n Subscript Baseline mathematical right-angle Subscript g should be considered as a single mathematical symbol, and the order of the d Subscript i ’s doesn’t matter.

Define the generating function for top intersection products in genus g by

upper F Subscript g Baseline left-parenthesis t 0 comma t 1 comma ellipsis right-parenthesis equals sigma-summation Underscript n Endscripts StartFraction 1 Over n factorial EndFraction sigma-summation Underscript d 1 comma ellipsis comma d Subscript n Baseline Endscripts mathematical left-angle product tau Subscript d Sub Subscript i Subscript Baseline mathematical right-angle Subscript g Baseline t Subscript d 1 Baseline midline-horizontal-ellipsis t Subscript d Sub Subscript n Subscript Baseline comma

where the sum is over all nonnegative sequences left-parenthesis d 1 comma ellipsis comma d Subscript n Baseline right-parenthesis such that sigma-summation d Subscript i Baseline equals 3 g minus 3 plus n . One can then form the generating function

upper F equals sigma-summation Underscript g Endscripts lamda Superscript 2 g minus 2 Baseline upper F Subscript g Baseline comma

which arises as a partition function in two-dimensional quantum gravity. Note that upper F is a generating function in infinitely many variables: lamda keeps track of the genus, and t Subscript d keeps track of the number of d th powers of psi classes.

Witten’s conjecture is equivalent to the fact that e Superscript upper F is annihilated by a sequence of differential operators

upper L Subscript negative 1 Baseline comma upper L 0 comma upper L 1 comma upper L 2 comma ellipsis

satisfying the Virasoro relations

left-bracket upper L Subscript m Baseline comma upper L Subscript k Baseline right-bracket equals left-parenthesis m minus k right-parenthesis upper L Subscript m plus k Baseline period

To give an idea of the complexity of these operators, we record the formula for upper L Subscript n Baseline comma n greater-than 0 :

StartLayout 1st Row 1st Column upper L Subscript n Baseline equals 2nd Column minus StartFraction left-parenthesis 2 n plus 3 right-parenthesis factorial factorial Over 2 Superscript n plus 1 Baseline EndFraction StartFraction partial-differential Over partial-differential t Subscript n plus 1 Baseline EndFraction 2nd Row 1st Column Blank 2nd Column plus StartFraction 1 Over 2 Superscript n plus 1 Baseline EndFraction sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts StartFraction left-parenthesis 2 k plus 2 n plus 1 right-parenthesis factorial factorial Over left-parenthesis 2 k minus 1 right-parenthesis factorial factorial EndFraction t Subscript k Baseline StartFraction partial-differential Over partial-differential t Subscript n plus k Baseline EndFraction 3rd Row 1st Column Blank 2nd Column plus StartFraction 1 Over 2 Superscript n plus 2 Baseline EndFraction sigma-summation Underscript i plus j equals n minus 1 Endscripts left-parenthesis 2 i plus 1 right-parenthesis factorial factorial left-parenthesis 2 k plus 1 right-parenthesis factorial factorial StartFraction partial-differential squared Over partial-differential t Subscript i Baseline partial-differential t Subscript j Baseline EndFraction period EndLayout

The equations upper L Subscript i Baseline left-parenthesis e Superscript upper F Baseline right-parenthesis equals 0 encode recursions among the intersection numbers, which appear as the constant terms in Mirzakhani’s volume polynomials. These recursions allow for the computation of all intersection numbers of psi classes. Mirzakhani showed that these recursions follow from her recursive formulas for the volume polynomials, thus giving a new proof of Witten’s conjecture.

8.3. A brief history

Witten’s conjecture was published in 1991, motivated by physical intuition that two different models for two-dimensional quantum gravity should be equivalent Reference Wit91. Kontsevich published a proof in 1992, using a combinatorial model for script upper M Subscript g comma n arising from Strebel differentials, ribbon graphs, and random matrices Reference Kon92. This work was central in his 1998 Fields Medal citation.

It was not until 2007 that Mirzakhani’s proof was published, and around the same time other proofs appeared. Later, Do related Mirzakhani’s and Kontsevich’s proofs, recovering Kontsevich’s formula for the number of ribbon graphs by considering asymptotics of the Weil–Petersson volume polynomials, using that a rescaled Riemann surface with very large geodesic boundary looks like a graph Reference Do10.

9. Counting simple closed geodesics

Let upper X be a complete hyperbolic surface, and let c Subscript upper X Baseline left-parenthesis upper L right-parenthesis be the number of primitive closed geodesics of length at most upper L on upper X . “Primitive” means simply that the geodesic does not traverse the same path multiple times. The famous Prime Number Theorem for Geodesics gives the asymptotic

c Subscript upper X Baseline left-parenthesis upper L right-parenthesis tilde one half StartFraction e Superscript upper L Baseline Over upper L EndFraction

as upper L right-arrow normal infinity . (The factor of one half disappears if one counts primitive oriented geodesics, since there are two orientations on each closed geodesic.) Amazingly, this does not depend on which surface upper X we choose or even the genus of upper X .

In Reference Mir08b, Mirzakhani proved that the number of closed geodesics of length at most upper L on upper X that do not intersect themselves is asymptotic to a constant depending on upper X times upper L Superscript 6 g minus 6 plus 2 n . That this asymptotic is polynomial rather than exponential reflects the extreme unlikeliness that a random closed geodesic is simple, in the same spirit as the result of Birman and Series mentioned in Section 5.

In fact Mirzakhani proved a more general result. For any rational multi-curve gamma , she considered

s Subscript upper X Baseline left-parenthesis upper L comma gamma right-parenthesis equals StartAbsoluteValue StartSet alpha element-of upper M upper C upper G dot gamma colon script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis less-than-or-equal-to upper L EndSet EndAbsoluteValue period

In other words, s Subscript upper X Baseline left-parenthesis upper L comma gamma right-parenthesis counts the number of closed multi-geodesics alpha on upper X of length less than upper L that are “of the same topological type” as gamma .

The set of simple closed curves forms finitely many mapping class group orbits. So by summing finitely many of these functions s Subscript upper X Baseline left-parenthesis upper L comma gamma right-parenthesis , one gets the corresponding count for all simple closed curves.

Theorem 9.1.

For any rational multi-curve gamma ,

limit Underscript upper L right-arrow normal infinity Endscripts StartFraction s Subscript upper X Baseline left-parenthesis upper L comma gamma right-parenthesis Over upper L Superscript 6 g minus 6 plus 2 n Baseline EndFraction equals StartFraction c left-parenthesis gamma right-parenthesis dot upper B left-parenthesis upper X right-parenthesis Over b Subscript g comma n Baseline EndFraction comma

where c left-parenthesis gamma right-parenthesis element-of double-struck upper Q Subscript plus , upper B colon script upper M Subscript g comma n Baseline right-arrow double-struck upper R Subscript plus Baseline is a proper, continuous function with a simple geometric definition, and b Subscript g comma n Baseline equals integral Underscript script upper M Subscript g comma n Baseline Endscripts upper B left-parenthesis upper X right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P Baseline .

Eskin, Mirzakhani, and Mohammadi have recently given a new proof of Theorem 9.1 that gives an error term, which we will discuss in Section 18, and Erlandsson and Souto have also given a new proof in Reference ES19. Here we outline the original proof, after first commenting on one application.

9.1. Relative frequencies

Consider, for example, the case left-parenthesis g comma n right-parenthesis equals left-parenthesis 2 comma 0 right-parenthesis of closed genus 2 surfaces, just to be concrete. The set of simple closed curves consists of two mapping class groups orbits: the orbit of a nonseparating curve gamma Subscript n s and the orbit of a curve gamma Subscript s e p that separates the surface into two genus 1 subsurfaces.

The fact that the limit in Theorem 9.1 is the product of a function of gamma and a function of upper X has the following consequence: A very long simple closed curve on upper X , chosen at random among all such curves, has probability about

StartFraction c left-parenthesis gamma Subscript s e p Baseline right-parenthesis Over c left-parenthesis gamma Subscript n s Baseline right-parenthesis plus c left-parenthesis gamma Subscript s e p Baseline right-parenthesis EndFraction

of being separating. Remarkably, this probability is computable and does not depend on upper X !

Even more remarkably, recent discoveries prove that the same probabilities appear in discrete problems about surfaces assembled out of finitely many unit squares Reference DGZZReference AH19.

9.2. The space of measured foliations

The space of rational multi-curves admits a natural completion called the space script upper M script upper F of measured foliations. Later we will delve into measured foliations, but here we only need a few properties of this space.

script upper M script upper F is homeomorphic to double-struck upper R Superscript 6 g minus 6 plus 2 n Baseline period

script upper M script upper F does not carry a natural linear structure. The most superficial indication of this is that any closed curve alpha gives a point of script upper M script upper F , but there is no negative alpha in script upper M script upper F , because multi-curves are defined to have positive coefficients. There is however a natural action of double-struck upper R Subscript plus on script upper M script upper F , which on multi-curves simply multiplies the coefficients by t element-of double-struck upper R Subscript plus .

script upper M script upper F has a natural piecewise linear integral structure, that is, an atlas of charts to double-struck upper R Superscript 6 g minus 6 plus 2 n whose transition functions are piecewise in normal upper G normal upper L left-parenthesis n comma double-struck upper Z right-parenthesis .

Define the integral points of script upper M script upper F as the set script upper M times script upper F left-parenthesis double-struck upper Z right-parenthesis subset-of script upper M script upper F of points mapping to double-struck upper Z Superscript 6 g minus 6 plus 2 n under the charts. Define the rational points script upper M times script upper F left-parenthesis double-struck upper Q right-parenthesis similarly. Then integral (resp., rational) points of script upper M script upper F parametrize homotopy classes of integral (resp., rational) multi-curves on the surface.

Any upper X element-of script upper M Subscript g comma n defines a continuous length function script upper M script upper F right-arrow double-struck upper R Subscript plus Baseline comma lamda right-arrow script l Subscript lamda Baseline left-parenthesis upper X right-parenthesis comma

whose restriction to multi-curves gives the hyperbolic length of the geodesic representative of the multi-curve on upper X . In particular, script l Subscript t lamda Baseline left-parenthesis upper X right-parenthesis equals t script l Subscript lamda Baseline left-parenthesis upper X right-parenthesis for all t element-of double-struck upper R .

9.3. Warmup

How many points of double-struck upper Z squared subset-of double-struck upper R squared have length at most upper L ? It is equivalent to ask about the number of points of StartFraction 1 Over upper L EndFraction double-struck upper Z squared contained in the unit ball.

Recall that the Lebesgue measure can be defined as the limit as upper L right-arrow normal infinity of

StartFraction 1 Over upper L squared EndFraction sigma-summation Underscript alpha element-of double-struck upper Z squared Endscripts delta Subscript StartFraction 1 Over upper L EndFraction alpha Baseline comma

where delta Subscript x denotes the point mass at x . Hence the number of points of StartFraction 1 Over upper L EndFraction double-struck upper Z squared contained in the unit ball is asymptotic to upper L squared times the Lebesgue measure of the unit ball.

9.4. The Thurston measure

Let us start with the easy question of asymptotics for the number

upper S Subscript upper X Baseline left-parenthesis upper L right-parenthesis equals StartAbsoluteValue StartSet alpha element-of script upper M times script upper F left-parenthesis double-struck upper Z right-parenthesis colon script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis less-than-or-equal-to upper L EndSet EndAbsoluteValue

of all integral multi-curves of length at most upper L . Using script l Subscript t lamda Baseline left-parenthesis upper X right-parenthesis equals t script l Subscript lamda Baseline left-parenthesis upper X right-parenthesis comma we observe that

upper S Subscript upper X Baseline left-parenthesis upper L right-parenthesis equals StartAbsoluteValue StartSet alpha element-of upper L Superscript negative 1 Baseline script upper M times script upper F left-parenthesis double-struck upper Z right-parenthesis colon script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis less-than-or-equal-to 1 EndSet EndAbsoluteValue period

It is now useful to define the “unit ball”

upper B Subscript upper X Baseline equals StartSet alpha element-of script upper M script upper F colon script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis less-than-or-equal-to 1 EndSet

and the measures

mu Superscript upper L Baseline equals StartFraction 1 Over upper L Superscript 6 g minus 6 plus 2 n Baseline EndFraction sigma-summation Underscript alpha element-of script upper M times script upper F left-parenthesis double-struck upper Z right-parenthesis Endscripts delta Subscript StartFraction 1 Over upper L EndFraction alpha Baseline period

With these definitions,

upper S Subscript upper X Baseline left-parenthesis upper L right-parenthesis equals upper L Superscript 6 g minus 6 plus 2 n Baseline mu Superscript upper L Baseline left-parenthesis upper B Subscript upper X Baseline right-parenthesis period

As in our warm up, the measures mu Superscript upper L converge to a natural Lebesgue class measure mu Subscript upper T h on script upper M script upper F . This measure mu Subscript upper T h is called the Thurston measure and is Lebesgue measure in the charts mentioned above. If we define upper B left-parenthesis upper X right-parenthesis equals mu Subscript upper T h Baseline left-parenthesis upper B Subscript upper X Baseline right-parenthesis , we get the asymptotic

upper S Subscript upper X Baseline left-parenthesis upper L right-parenthesis tilde upper B left-parenthesis upper X right-parenthesis upper L Superscript 6 g minus 6 plus 2 n Baseline period

9.5. The proof

Mirzakhani’s approach to Theorem 9.1 similarly defines measures

mu Subscript gamma Superscript upper L Baseline equals StartFraction 1 Over upper L Superscript 6 g minus 6 plus 2 n Baseline EndFraction sigma-summation Underscript alpha element-of upper M upper C upper G dot gamma Endscripts delta Subscript StartFraction 1 Over upper L EndFraction alpha Baseline period

As above, to prove Theorem 9.1, it suffices to show the convergence of measures

mu Subscript gamma Superscript upper L Baseline right-arrow StartFraction c left-parenthesis gamma right-parenthesis Over b Subscript g comma n Baseline EndFraction mu Subscript upper T h Baseline period

Using the Banach–Alaoglu Theorem, it is not hard to show that there are subsequences upper L Subscript i Baseline right-arrow normal infinity such that mu Subscript gamma Superscript upper L Super Subscript i converges to some measure mu Subscript gamma Superscript normal infinity , which might a priori depend on which subsequence we pick. To prove Theorem 9.1 it suffices to show that, no matter which such subsequence we use, we have

mu Subscript gamma Superscript normal infinity Baseline equals StartFraction c left-parenthesis gamma right-parenthesis Over b Subscript g comma n Baseline EndFraction mu Subscript upper T h Baseline period

By definition, mu Subscript gamma Superscript upper L Baseline less-than-or-equal-to mu Superscript upper L , since the mapping class group orbit of gamma is a subset of script upper M times script upper F left-parenthesis double-struck upper Z right-parenthesis . Since mu Superscript upper L converges to the Thurston measure, we get that mu Subscript gamma Superscript normal infinity Baseline less-than-or-equal-to mu Subscript upper T h .

Since mu Subscript gamma Superscript upper L is mapping class group invariant, the same is true for mu Subscript gamma Superscript normal infinity . A result of Masur in ergodic theory, which we will discuss in Section 13, gives that any mapping class group invariant measure on script upper M script upper F that is absolutely continuous to mu Subscript upper T h must be a multiple of mu Subscript upper T h Reference Mas85.

So mu Subscript gamma Superscript upper L Baseline less-than-or-equal-to mu Superscript upper L Baseline equals c mu Subscript upper T h for some c greater-than-or-equal-to 0 . At this point in the argument, as far as we know, c could depend on the subsequence of upper L Subscript i .

Unraveling the definitions, we have that

StartLayout 1st Row with Label left-parenthesis 9.5 .1 right-parenthesis EndLabel StartFraction s Subscript upper X Baseline left-parenthesis upper L Subscript i Baseline comma gamma right-parenthesis Over upper L Subscript i Superscript 6 g minus 6 plus 2 n Baseline EndFraction right-arrow c dot upper B left-parenthesis upper X right-parenthesis comma EndLayout

for any upper X element-of script upper M Subscript g comma n . Writing

s Subscript upper X Baseline left-parenthesis upper L Subscript i Baseline comma gamma right-parenthesis equals sigma-summation Underscript alpha element-of upper M upper C upper G gamma Endscripts chi Subscript left-bracket 0 comma upper L Sub Subscript i Subscript right-bracket Baseline left-parenthesis script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis right-parenthesis comma

we recognize the type of function that Mirzakhani’s Integration Formula applies to. By integrating the left-hand side of Equation 9.5.1 over moduli space, Mirzakhani is able to prove that c equals StartFraction c left-parenthesis gamma right-parenthesis Over b Subscript g comma n Baseline EndFraction as desired. On the one hand, the integral of c upper B left-parenthesis upper X right-parenthesis is c dot b Subscript g comma n . On the other hand, the limit c left-parenthesis gamma right-parenthesis of the integral of StartFraction s Subscript upper X Baseline left-parenthesis upper L Subscript i Baseline comma gamma right-parenthesis Over upper L Subscript i Superscript 6 g minus 6 plus 2 n Baseline EndFraction is easily expressed in terms of the leading order term in one of Mirzakhani’s volume polynomials.

9.6. Open problems

We will return to counting later, but for now we mention the following.

Problem 9.2.

Prove an analogue of Theorem 9.1 for nonorientable hyperbolic surfaces.

An example is known already with asymptotics upper L Superscript delta with delta nonintegral Reference Mag17. See Reference Gen17 for a more precise conjecture, as well as a number of related open problems and an analogy between moduli spaces of nonoriented hyperbolic surfaces and infinite volume geometrically finite hyperbolic manifolds.

10. Random surfaces of large genus

Given a random d -regular graph with many vertices, what is the chance that it contains a short loop? Is a random graph easy to cut in two? What properties can be expected of the graph Laplacian?

Mirzakhani considered analogues of these well-studied questions for Weil–Petersson random Riemann surfaces Reference Mir13Reference MZ15Reference MP17 and devoted her 2010 talk at the International Congress of Mathematicians to this topic Reference Mir10.

In this section we discuss this work. We will leave out the background on graphs, but many readers will wish to keep in mind the comparison between a random d -regular graph, with d fixed and a large number of vertices, and a random surface with large genus.

10.1. Understanding the volume polynomials

We begin with the constant term of the polynomial upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis right-parenthesis , which is the volume upper V Subscript g comma n of script upper M Subscript g comma n . Improving on previous results of Mirzakhani and others, Mirzakhani and Zograf proved the following Reference Mir13Reference MZ15.

Theorem 10.1.

There exists a universal constant upper C element-of left-parenthesis 0 comma normal infinity right-parenthesis such that for any fixed n , upper V Subscript g comma n is asymptotic to

upper C StartFraction left-parenthesis 2 g minus 3 plus n right-parenthesis factorial left-parenthesis 4 pi squared right-parenthesis Superscript 2 g minus 3 plus n Baseline Over StartRoot g EndRoot EndFraction

as g right-arrow normal infinity .

This largely verified a previous conjecture of Zograf, except that his prediction that upper C equals StartFraction 1 Over StartRoot pi EndRoot EndFraction is still open Reference Zog08. Mirzakhani and Zograf also gave a more detailed asymptotic expansion. The proof uses the recursions satisfied by upper V Subscript g comma n discussed in Section 6.

Previous results gave asymptotics as n right-arrow normal infinity for fixed g Reference MZ00. See Reference Mir13, Section 1.4 for open questions concerning asymptotics as both g and n go to infinity.

Also by studying recursions, Mirzakhani proved results in Reference Mir13 that imply

StartLayout 1st Row with Label left-parenthesis 10.1 .1 right-parenthesis EndLabel upper V o l left-parenthesis script upper M Subscript g comma n Baseline left-parenthesis upper L right-parenthesis right-parenthesis less-than-or-equal-to upper V Subscript g comma n Baseline product Underscript i equals 1 Overscript n Endscripts StartFraction hyperbolic sine left-parenthesis upper L Subscript i Baseline slash 2 right-parenthesis Over upper L Subscript i Baseline slash 2 EndFraction period EndLayout

Mirzakhani and Petri showed this bound is asymptotically sharp for fixed n and bounded upper L as g right-arrow normal infinity Reference MP17, Proposition 3.1. The proof of the inequality actually gives a bound with hyperbolic sine replaced with one of its Taylor polynomials.

10.2. An example

To illustrate Mirzakhani’s techniques, we will give an upper bound for the probability that a random surface in script upper M Subscript g has a nonseparating simple closed geodesic of length at most some small epsilon greater-than 0 .

We begin by studying the average over script upper M Subscript g of the number of simple, nonseparating geodesics of length at most epsilon on upper X element-of script upper M Subscript g . If gamma is a simple nonseparating curve, we can express this as

StartFraction 1 Over upper V Subscript g Baseline EndFraction integral Underscript script upper M Subscript g Baseline Endscripts sigma-summation Underscript alpha element-of upper M upper C upper G dot gamma Endscripts chi Subscript left-bracket 0 comma epsilon right-bracket Baseline left-parenthesis script l Subscript alpha Baseline left-parenthesis upper X right-parenthesis right-parenthesis normal d normal upper V normal o normal l Subscript upper W upper P Baseline comma

where chi Subscript left-bracket 0 comma epsilon right-bracket is the characteristic function of the interval left-bracket 0 comma epsilon right-bracket . Mirzakhani’s Integration Formula gives that this is equal to a constant times

StartFraction 1 Over upper V Subscript g Baseline EndFraction integral Subscript 0 Superscript epsilon Baseline script l upper V o l left-parenthesis script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis right-parenthesis normal d script l period

Since script l is small, inequality Equation 10.1.1 gives that upper V o l left-parenthesis script upper M Subscript g minus 1 comma 2 Baseline left-parenthesis script l comma script l right-parenthesis right-parenthesis is approximately equal to the constant term upper V Subscript g minus 1 comma 2 of the volume polynomial, so the average is approximately a constant times

StartFraction upper V Subscript g minus 1 comma 2 Baseline Over upper V Subscript g Baseline EndFraction epsilon squared period

The asymptotics in Theorem 10.1 imply that StartFraction upper V Subscript g minus 1 comma 2 Baseline Over upper V Subscript g Baseline EndFraction converges to 1 as g right-arrow normal infinity , so we get that the average number of simple, nonseparating geodesics of length at most epsilon is asymptotic, as g right-arrow normal infinity , to a constant times epsilon squared . In particular, this implies that the probability that a random surface in script upper M Subscript g has such a geodesic is bounded above by a constant times epsilon squared .

A similar lower bound is possible by giving upper bounds for the average number of pairs of nonseparating simple closed curves.

10.3. Results

Here is an overview of results from Reference Mir13, which concern random upper X element-of script upper M Subscript g as g right-arrow normal infinity .

The probability that upper X has a geodesic of length at most epsilon is bounded above and below by a constant times epsilon squared .

The probability that upper X has a separating geodesic of length at most 1.99 log left-parenthesis g right-parenthesis goes to 0.

The probability that upper X has Cheeger constant less than 0.099 goes to 0.

The probability that lamda 1 left-parenthesis upper X right-parenthesis , the first eigenvalue of the Laplacian, is less than 0.002 goes to 0.

The probability that the diameter of upper X is greater than 40 log left-parenthesis g right-parenthesis goes to 0.

The probability that upper X has an embedded ball of radius at least log left-parenthesis g right-parenthesis slash 6 goes to 1.

The first two results are proven using the techniques in the example. The Cheeger constant is defined as

h left-parenthesis upper X right-parenthesis equals inf Underscript alpha Endscripts StartFraction script l left-parenthesis alpha right-parenthesis Over min left-parenthesis upper A r e a left-parenthesis upper X 1 right-parenthesis comma upper A r e a left-parenthesis upper X 2 right-parenthesis right-parenthesis EndFraction comma

where the infimum is over all smooth multi-curves alpha that cut upper X into two subsurfaces upper X 1 comma upper X 2 . Mirzakhani defines the geodesic Cheeger constant upper H left-parenthesis upper X right-parenthesis to be the same quantity where alpha is required to be a geodesic multi-curve, so obviously h left-parenthesis upper X right-parenthesis less-than-or-equal-to upper H left-parenthesis upper X right-parenthesis . She proves that

StartFraction upper H left-parenthesis upper X right-parenthesis Over upper H left-parenthesis upper X right-parenthesis plus 1 EndFraction less-than-or-equal-to h left-parenthesis upper X right-parenthesis comma

and is then able to study upper H left-parenthesis upper X right-parenthesis using the techniques in the example. The result on lamda 1 follows from the Cheeger inequality lamda 1 greater-than-or-equal-to h left-parenthesis upper X right-parenthesis squared slash 4 .

We conclude with a special case of the main result of Mirzakhani and Petri Reference MP17.

Theorem 10.2.

For any 0 less-than a less-than b , the number of primitive closed geodesics of length in left-bracket a comma b right-bracket , viewed as a random variable on script upper M Subscript g , converges to a Poisson distribution as g right-arrow normal infinity .

What is fascinating about this result of Mirzakhani and Petri is that it concerns all primitive closed geodesics, not just the simple ones. The proof uses that a geodesic gamma of length at most a constant b on a surface upper X of very large genus is contained in a subsurface of bounded genus and with a bounded number of boundary components (depending on b ). The boundary of that subsurface is a simple multi-curve beta associated to gamma . By showing that, as g right-arrow normal infinity , most upper X do not have a separating multi-curve of bounded length, they are able to show that on most upper X most primitive geodesics are simple and, hence, use the techniques illustrated in the example.

10.4. Open problems

For some problems, we list an easier version followed by a harder version.

Problem 10.3.

Does there exist a sequence of Riemann surfaces upper X Subscript n