PDFLINK |

# Counting Geodesics, Teichmüller Space, and Random Hyperbolic Surfaces

Communicated by *Notices* Associate Editor Chikako Mese

## Introduction

The study of geodesics provides a theme for understanding hyperbolic metrics on finite-area surfaces, as well as the geometry of the moduli space of Riemann surfaces. We begin with the fundamentals of the Teichmüller theory of hyperbolic surfaces. We relate how formulas in hyperbolic geometry correspond to formulas for the Teichmüller space and moduli space. Then we describe how Thurston’s random geodesic metric generalizes to the pressure metric in higher Teichmüller theory and how Mirzakhani’s recursive integration scheme is a tool for understanding random finite-area hyperbolic surfaces. Our informal exposition does not attempt a summary of a historical development of the subject. Too many important results are not discussed and the development of ideas is greatly simplified. Only limited references are provided. For further resources the reader may explore McMullen’s Expositions and Course Notes at the bottom of his Publications. A general introduction to the present material is provided in “Riemann surfaces, dynamics and geometry”—all available on people.math.harvard.edu/~ctm/. The reader may also explore the author’s CBMS and PCMI lectures 1112. And a valuable exercise is to use AMS MATHSCINET to check the publications that refer to the references here—a reverse search.

## Hyperbolic Geometry and Riemann Surfaces

We start with the model for the hyperbolic plane provided by the upper half-plane with the Riemannian metric 35. The group of orientation-preserving isometries is given by acting by fractional linear transformations on preserving the upper half-plane. The maximal geodesics are the semicircles with centers on the real axis and the vertical half-lines. A *Riemann surface* is prescribed by a covering of a topological space by an atlas of charts with maps to the complex plane such that the overlap maps are biholomorphic. A Riemannian metric for a surface is prescribed by specifying positive coefficients such that Alternately, a hyperbolic metric can be specified by special .charts— with maps to the hyperbolic plane with local isometries. A surface with metric is also described by gluing (attaching by isometries) shapes from a model space. For example a flat torus is given by identifying opposite sides of a Euclidean parallelogram.

The classical *modular surface* is given by gluing the domain The hyperbolic triangle . has vertices at and , and an ideal vertex with zero angle at infinity. The vertical sides are glued by the translation , The semicircle segment . is glued to the semicircle segment by the negative inversion A neighborhood of the image of . in the quotient space is alternately described by with identifying the semicircles and The points . and correspond to a cone point in the quotient space. The total angle around the point is A neighborhood of the image of infinity in the quotient space is a cusp—the point infinity is at infinite distance. The area of . is the angle defect The matrices . and generate the classical *modular group* quotient of the group of —the integer matrices with determinant by The domain . is a fundamental domain for the action of the modular group on the hyperbolic plane. The translates of exactly tile and the quotient , is the identification space of .

Convexity is a basic feature of hyperbolic geometry. Accordingly a closed loop on a hyperbolic surface can be pulled taut to form a closed geodesic. Each nontrivial, noncuspidal, free homotopy class contains a unique closed geodesic that is length minimizing. Similarly, each free homotopy class of arcs (with sliding endpoints) between a pair of closed geodesics contains a unique geodesic segment. The segment realizes the distance. Geodesics in hyperbolic geometry exhibit very special behaviors. Following a result of Birman-Series, Sapir showed that on a finite-area surface the collection of geodesics with bounds on the number of self-intersections is contained in a closed set of Hausdorff dimension one and Lebesgue measure zero. Geodesics with less than maximal self-intersections lie only in narrow channels on a surface. By total contrast, the collection of all closed possibly self-intersecting geodesics is *uniformly distributed*. Given small, there is an large such that given a tangent vector the total length of segments of closed geodesics of length at most , with tangents to -close is positive and approximately independent of In fact, after normalization the total length of segments close to a given tangent vector tends to the uniform distribution in . as tends to infinity. In effect closed geodesics almost uniformly almost pass through each point with almost each tangent direction. Consequently we can contemplate the behavior of the *random closed geodesic*. A starter question is to count by length the number of closed geodesics. In analogy to the prime number theorem, Delsarte, Huber, and Selberg each introduced a trace formula to perform a count. For a finite-area hyperbolic surface the count is exponential. The number of closed geodesics of length at most is asymptotic to We will see that the count is polynomial when the number of self-intersections is restricted. .

By the Uniformization Theorem, a surface with complete hyperbolic metric is described as a quotient where , is a subgroup of The group . is isomorphic to the fundamental group of and acts on as the group of deck transformations. The hyperbolic plane is tiled by of a fundamental domain -translates a finite-sided polygon. If , has genus and cusps, the Euler characteristic is negative and the area of is .

## Teichmüller Spaces

Fenchel-Nielsen introduced a synthetic geometric tailoring construction for prescribing a hyperbolic surface 3511. The construction begins with the observation that in the hyperbolic plane right-angled hexagons are specified by giving the lengths of alternate sides as positive real numbers. Side lengths can also have the ideal value zero and then the adjacent sides of the hexagon describe a cusp as for the fundamental domain. Given a right hexagon with side lengths and , then double , across the alternate sides to obtain a pair of pants surface with the topology of a sphere minus three disjoint discs. The pants —a has geodesic boundaries since the double across one boundary of a quarter-plane is a half-plane. The boundary lengths are and , and the area is Also attaching half-planes is a local construction. Accordingly pants . and with a common boundary length can be attached across the common boundary to form a hyperbolic surface with four geodesic boundaries and the topology of a sphere minus four disjoint discs. Remaining boundaries of equal length can also be identified. For example starting with two copies of a pair of pants the boundaries of equal lengths can be identified in pairs to form a compact surface of genus , In general a surface of genus . with cusps is tailored by starting with pairs of pants and identifying in pairs boundaries of equal lengths. The collection of seams, a collection of disjoint simple (no self-intersections) closed geodesics, is called a *pants decomposition*.

What are the parameters for constructing a surface by tailoring pants? First there are the seam lengths, values in Second there is a twist parameter for each seam. A pair of pants has an equator reflection given by interchanging hexagons. The fixed set of a reflection is the three geodesic segments between boundary components. Since boundaries are circles, pants can be assembled with an arbitrary relative displacement between boundaries. The displacement between the equators on the two sides of a seam is a parameter. The Fenchel-Nielsen .*twist parameter* is the displacement between equators measured in hyperbolic length units. The twist parameter similar to angle measure takes values in Tailoring includes labeling pants and loops—the instructions also prescribe a basis for the fundamental group of a surface. Teichmüller space is the space of hyperbolic surfaces with a reference isomorphism for the fundamental groups modulo an equivalence relation. .

Different pants decompositions provide different global coordinates. In the next section we see that a completion of Teichmüller space is prescribed by allowing collections of pants lengths to vanish.

How many ways can an individual surface be tailored? A simple redundancy is given by adding the seam length to a twist parameter This parameter alteration corresponds to applying a homeomorphism, a Dehn twist, that is a full twist in a neighborhood of the seam curve and the identity otherwise. More generally new pants decompositions for a fixed surface are given by applying homeomorphisms. Even more generally there are pants decompositions not related by homeomorphisms. For example in genus two, modulo homeomorphisms, there are two distinct decompositions: one with no separating curves and one with a single separating curve. Bollobás found the asymptotic count of distinct pants decompositions—the leading term is . By definition decompositions related by homeomorphisms are related by the mapping class group . group of orientation-preserving homeomorphisms of a surface modulo its subgroup of homeomorphisms isotopic to the identity. The group acts transitively on the reference isomorphisms for the points of the Teichmüller space —the The action of . on is properly discontinuous. By a theorem of Dehn, is generated by Dehn twists. The quotient is the *moduli space* of Riemann surfaces—the space of distinct Riemann surfaces.

## Moduli Spaces

A Fenchel-Nielsen tailoring description of is given by a description of an fundamental domain in By all indications a general description is expected to be intractable. For simple applications an estimate of Buser and Parlier suffices. Every closed hyperbolic surface has a pants decomposition with all seams of length at most . .

A basic deformation of hyperbolic surfaces is pinching collars—a taffy pulling deformation. Given a simple closed geodesic of length the ,*collar* about is the tubular neighborhood for with the hyperbolic metric on isometric to The collar lemma provides that simple geodesics have collars and disjoint simple geodesics have disjoint collars. Since . is large for small, it follows that short simple geodesics are disjoint. From an additional observation it follows that short geodesics are necessarily simple. The diameter of a containing surface is at least the collar half-width Taffy snaps into two pieces after extreme pulling. The limit of collars with core lengths . tending to zero is a pair of cusps. What are the possible limits for a sequence of hyperbolic surfaces? The first consideration is whether there is a positive lower bound for the lengths of closed geodesics.

The result can be extended. If collars about short geodesics are removed from hyperbolic surfaces, then the collection of metric spaces is precompact in the Gromov-Hausdorff topology. When pulling taffy, the ends deform only a bounded amount. The complements in hyperbolic surfaces of the collars about short geodesics only vary a bounded amount.

The considerations indicate that is compactified by simply permitting collars to limit to pairs of cusps. In particular given Fenchel-Nielsen coordinates a ,*boundary space* is added to Teichmüller space by permitting length parameters to take the describing cusps for the corresponding pants and hexagons. The twist parameter is undefined for zero length. Paired cusps are always included as elements of a pants decomposition. A surface with paired cusps and a fundamental group reference isomorphism is described by multiple pants decompositions—giving rise to an equivalence relation for boundary spaces. -value,

The mapping class group acts on and the quotient ,

## Weil-Petersson Geometry

How do we describe the geometry of Teichmüller space *twist vector field* *geodesic-length function*

We begin with Riera’s formula for the pairing of geodesic-length gradients—the formula can serve as a definition for the pairing. Introduce the positive function

An analysis involving the geometry of collars shows for

There is a rich symplectic geometry of twists and lengths corresponding to the hyperbolic geometry of surfaces. The WP metric is Kähler with symplectic form

The cosine formula is valid even when the Teichmüller space is a single point. In particular the sum of cosines is zero for any two closed geodesics on the modular surface

This simple formula is an ingredient in Mirzakhani’s integration scheme.

Convexity is also a consideration for understanding WP geodesics.

Yamada combined geodesic convexity and an analysis of the metric near boundary spaces to show that

## The Random Geodesic Metric

The *geodesic flow* on the set of unit tangent vectors of a manifold is defined by following a geodesic with a given initial tangent. For negatively curved manifolds the flow is topologically transitive—has a dense orbit. The flow is also Anosov—the bundle of unit tangent vectors decomposes into three invariant subspaces: one on which the flow is expanding, one on which the negative time flow is expanding, and the one-dimensional bundle of tangents to the flow. The closed orbits of the flow are the closed geodesics with their tangents. For hyperbolic surfaces Thurston observed that the statistics of variations of geodesic lengths define a metric on Teichmüller space. More recently from the work of a collection of authors, we know the Thermodynamic Formalism can be used to measure the difference of Anosov-type flows and gives rise to a seminorm, the pressure metric, on certain deformation spaces.

Thurston’s definition is based on the properties of uniformly distributed orbits and on properties of twist derivatives.

Observations are needed. Uniform distribution provides that the number of intersections

The trigonometric expressions for the second twist derivative converge to the trigonometric expressions in Riera’s formula.

The Thermodynamic Formalism provides a general framework for using the statistics of periods of closed orbits to define a seminorm 12. Let

Bowen observed that the *topological entropy* is the exponential growth rate of the count:

If

Bowen-Ruelle and Sambarino consider the *pressure* of

The above period integral is the model for defining a modified time scale for the flow. Given a positive Hölder function *reparameterized* flow with periods

The integral of

In this setting Ruelle showed that the pressure is a convex function—in particular for Hölder functions

Sambarino showed that the topological entropy is precisely the scale factor to give pressure zero—in particular for

Following McMullen a pressure seminorm is defined on the space of pressure zero Hölder functions. In particular for *pressure seminorm* at

Ruelle and Parry-Pollicott showed that the pressure seminorm characterizes the variations of orbit periods.

Establishing the relation to Thurston’s random geodesic metric involves an intersection number introduced by Bonahon. The *renormalized intersection number* for two positive Hölder functions is

The Thermodynamic Formalism is suited for considering representation spaces. Teichmüller space is alternately described as the space of discrete faithful representations of a compact surface fundamental group into

Bridgeman-Canary-Labourie-Sambarino showed that there is an analytic mapping from

Bridgeman-Canary-Labourie-Sambarino developed the Thermodynamic Formalism for Hitchin representations into

Each Hitchin component contains an image of *Fuchsian locus*, induced by the mapping

Labourie developed an in-depth geometric description of Hitchin representations by analyzing the group actions on an appropriate limit set. Sambarino and then more generally Bridgeman-Canary-Labourie-Sambarino defined a topologically transitive Anosov-type flow with closed orbit periods given by logarithms of the maximal eigenvalues of

## The Mirzakhani Recursion

Mirzakhani answered how to integrate over the moduli space 6. The method begins with a generalization of McShane’s *length identity*. Then start with the identity and replace the sums over topological types with sums over translates of an

We consider surfaces with geodesic boundaries. Introduce the elementary functions

and

For a hyperbolic genus one surface

The boundary length has an exact relation to the lengths of all the simple closed geodesics. The identity comes from an analysis of geodesics perpendicular to the boundary. For the sake of exposition, we overlook the role of the elliptic involution in the following. The set of free homotopy classes of simple closed curves on a torus forms a single

So we write McShane’s identity as

Writing

writing