# Euclidean Traveller in Hyperbolic Worlds

Hee Oh

We will discuss all possible closures of a Euclidean line in various geometric spaces. Imagine the Euclidean traveller, who travels only along a Euclidean line. She will be traveling to many different geometric worlds, and our question will be what places does she get to see in each world?

Here is the itinerary of our Euclidean traveller:

In 1884, she travels to the torus of dimension , guided by Kronecker.

In 1936, she travels to the world, called a closed hyperbolic surface, guided by Hedlund Hed36.

In 1991, she then travels to a closed hyperbolic manifold of higher dimension guided by Ratner Rat91.

Finally, she adventures into hyperbolic manifolds of infinite volume guided by Dal’bo Dal00 in dimension two in 2000, by McMullen-Mohammadi-O. MMO16 in dimension three in 2016 and by Lee-O. LO19 in all higher dimensions in 2019.

## Rotations of the Circle

As a warm up, she will first do her exercise of jumping on the circle , which may be considered as the one-dimensional torus .

The circle can be presented as

These two models are isomorphic to each other by the logarithm map .

The map is respectively given by

in multiplicative and additive models of . The orbit of under iterations of is respectively equal to

If our traveller keeps jumping by an irrational distance , she is guaranteed to see all the places in the circle .

## Euclidean Lines on the Torus

### $2$-torus

She travels to the -torus . Let

denote the canonical quotient map. A Euclidean line in is the image of a line in under . For a nonzero vector , we denote by

the image of the unique line passing through and the origin . The slope of the line is equal to .

What is the closure of the line in , or in other words, what places does our Euclidean traveller get to see in if she travels along the line ?

It is useful to consider the unit square which is a fundamental domain of . When we identify the pairs of opposite sides labelled by and in Figure 2 using the side pairing transformations and respectively, we get the torus where is the subgroup generated by and . The distribution of the line can be understood by examining the line inside modulo the action of . In Figure 2, when our traveller, walking on the blue line with slope , reaches the boundary of at , she gets instantly jumped to by the transformation , which also moves the line to the line with the same slope but passing through . She then continues to walk on this new line in until she reaches the boundary of . We can observe that the places on the circle that she is visiting are precisely given by the orbit of the rotation . Therefore by Theorem 1, if the slope is a rational number, she visits only finitely many points in , which means the line is periodic and hence closed in . Otherwise, the places she visits in are dense, which means that the line is dense in .

### $n$-torus

For any , the -torus can be presented as , and a line in is the image of a line under the quotient map

For , let denote the line in which is the image of the Euclidean line .

Unlike the dimension two case, the closed or dense dichotomy for a line is not true anymore in , , as contains many lower dimensional tori. In Figure 3, the blue line is contained in a two-dimensional linear subspace such that is closed in and . However, the lower dimensional tori are the only other possibilities (cf. KH95):

A general Euclidean torus is defined as the quotient

for some discrete cocompact subgroup of ; a discrete subgroup of is called cocompact if the quotient space is compact. It is not hard to prove that any discrete cocompact subgroup of is of the form

for some basis , …, of .

This seemingly more general theorem follows easily from Theorem 3 using the fact that for any and where is an element of whose row vectors are given by , …, .

## Closed Hyperbolic Surfaces

Our Euclidean traveller now wants to explore a world called a closed hyperbolic surface. A closed hyperbolic surface will be defined as a quotient of the hyperbolic plane .

### Hyperbolic plane

The hyperbolic plane is the unique simply connected two-dimensional complete Riemannian manifold of constant sectional curvature . Instead of this fancy description, we will be using a very explicit model, called the Poincaré upper half-plane model of . That is,

with the hyperbolic metric given by . This means that the hyperbolic distance between is defined as

where , , ranges over all differentiable curves with and . Because the hyperbolic distance is the Euclidean distance scaled by the Euclidean height of the -coordinate, the hyperbolic distance between two points in is larger (resp. smaller) than their Euclidean distance if their Euclidean height is small (resp. large).

Geodesics, that is, distance-minimizing curves, in this upper half-plane model are half-lines or semicircles, perpendicular to the -axis. In other words, to travel from a point to in the fastest way, one has to take the route given by the unique circle (or a line) passing through and , perpendicular to the -axis.

Another useful model is the Poincaré unit disk model in which is the open unit disk in and the hyperbolic metric is given by .

Recall that

a Euclidean -torus is given by a quotient

where is a discrete cocompact subgroup of .

In analogy, we wish to be able to say that

a closed hyperbolic surface is given by a quotient

where is a discrete cocompact subgroup of .

Alas, the hyperbolic plane is not a group which makes this statement nonsense. However, is “almost” the same as the group of orientation preserving isometries of .

### Isometry group of $\mathbb{H}^2$

On the upper half-plane model , now considered as the set of complex numbers with positive imaginary parts, the group acts by linear fractional transformations:

since , this action preserves . We can also check that this action preserves the hyperbolic metric of . Therefore every element of is an isometry of . Moreover, it turns out that every orientation preserving isometry arises in this manner, yielding the identification

It is easy to see that this action of on is transitive with the stabilizer of being equal to the rotation group . Therefore the orbit map , , induces the identification

So modulo the compact subgroup , the hyperbolic plane is equal to its isometry group .

### Closed hyperbolic surfaces

By the quotient , we mean the set of equivalence classes of elements of where if and only if for some . The discreteness of implies that is locally and the cocompactness of in implies that is compact.

Does there exist a closed hyperbolic surface? Equivalently, does there exist a discrete cocompact subgroup of ? To present an example, consider the hyperbolic regular octagon as illustrated in Figure 5, where each side is a hyperbolic geodesic segment of same length and angles between them are .

For the two sides of the octagon labelled by , there exists an isometry which moves one to the other. Similarly, we have for labels respectively. Let

be the subgroup generated by these four side-pairing transformations. Then the hyperbolic octagon is a fundamental domain for the action of in , which implies that is a discrete cocompact subgroup of . The closed hyperbolic surface is what we obtain by gluing the four pairs of edges of the hyperbolic octagon according to labels; it is topologically a two-holed torus, or genus-two surface.

Any closed hyperbolic surface is topologically a closed surface with genus at least . Conversely, the Uniformization theorem says that any closed surface with genus at least can be realized as a closed hyperbolic surface. Indeed, for any , the space of all marked closed hyperbolic surfaces of genus is homeomorphic to .

Now that our traveller learned that there exist many (even a continuous family of) closed hyperbolic surfaces to explore, she has to understand what a Euclidean line is in this world.

## Euclidean Lines in Closed Hyperbolic Surfaces

In the upper half-plane model of , the horizontal line will be called a Euclidean line in . In a given geometric space, objects which are isometric to each other should be given the same name. We note that the image of under an isometry of is either another horizontal line, or a circle which is tangent to the -axis. So all of these objects will be called Euclidean lines in . A nickname for a Euclidean line is a horocycle. A horocycle in is characterized as an isometric embedding of the real line to with constant curvature one; geodesics in have constant curvature zero.

From the point of view of our Euclidean traveller, imagine that she wants to drive in a car where the steering wheel is in one fixed position and to travel without bumps. If her car is turning at a constant rate, she can lean back against the seat, feeling stable. It’s the change in curvature that makes the car trip bumpy. If the wheel is fixed at a small angle, she stays within a bounded distance of a geodesic. If the wheel is fixed at a large angle, she goes around in a circle and stays a bounded distance from a point. But in between there is a perfect angle where neither happens, and she moves along a horocycle!

A Euclidean line in a closed hyperbolic surface is the image of a Euclidean line in under the quotient map

Where does our Euclidean traveller get to visit in a closed hyperbolic surface?

Here is an illustration given in Figure 8. Fix a (blue-colored) fundamental domain for . When the traveller, walking along the mint-colored Euclidean line , reaches the boundary of , she gets instantly moved to a different side of by some side-pairing transformation . She then continues her journey on the new Euclidean line inside until she reaches the boundary of again, etc. The instant jumps and the shapes of translates of made by in appear more complicated than those in the Euclidean torus .

Nevertheless, Hedlund (1936) assures that our Euclidean traveller gets to see all the places in a closed hyperbolic surface, no matter where her initial point of departure is.

### Hyperbolic lines can be very wild

We remark that this theorem of Hedlund is about Euclidean lines. The closure of a hyperbolic line does not even have to be a submanifold in general.

As illustrated in Figure 10, a geodesic can “spiral” around closed geodesics and its closure can be a fractal of dimension strictly between one and two.

### Going upstairs

Hedlund’s proof of Theorem 5 relies on the fact that is almost . The isometric action of on , which gave us the identification , extends to an action on the unit tangent bundle . In this action, the stabilizer of a vector is trivial, as no rotation in the plane fixes a vector. If we denote the upward normal vector based at , then the orbit map now gives an isomorphism , where the identity matrix of corresponds to the vector . Moreover, if we consider the following one-dimensional subgroup

then the subgroup corresponds to the set of all upward normal vectors on the Euclidean line (see Figure 11).

Similarly, for with a circle tangent at , the coset corresponds to the set of all inward normal vectors on pointing to . Indeed, any Euclidean line (EL) in arises as the image of some under the basepoint projection map given by :

Since the image of in , under the projection , is equal to and , this picture is preserved under the quotient map :

It follows that any Euclidean line in is of the form and , as the basepoint projection map has compact fibers. Therefore if we can describe the closures of all orbits in , we understand the closure of a Euclidean line in .

Indeed, Hedlund says that every -orbit is dense upstairs in :

This means that our Euclidean traveller in a closed hyperbolic surface not only gets to see all the places, but she is able to appreciate those places from all angles as in Figure 9.

## Closed Hyperbolic $n$-manifolds

For , the hyperbolic -space is the unique simply connected complete -dimensional Riemannian manifold of constant sectional curvature . Its upper half space model is given by

with the hyperbolic metric . In this model, geodesics are vertical lines or semicircles meeting the hyperplane orthogonally.

The hyperboloid model (also called the Minkowski model) of is given by where

In this model, the hyperbolic distance is given by (cf. Rat94). In particular, any element of the special orthogonal group , that is, satisfying

is an isometry of . Indeed, the group of orientation preserving isometries is given by the identity component of the special orthogonal group ; so

This is consistent with our previous statement that , since . As in the dimension two case, we have

any closed hyperbolic -dimensional manifold is a quotient

where is a discrete (torsion-free) cocompact subgroup of .

Unlike the dimension two case where there is a continuous family of closed hyperbolic surfaces, higher dimensional closed hyperbolic manifolds are rarer. The Mostow rigidity theorem Mos73 implies that there exist only countably many closed hyperbolic manifolds of dimension at least three. Nevertheless, there are infinitely many of them Bor63.

In the upper half-space model , the horizontal line , and its isometric images will be called a Euclidean line (=horocycle) in . As before, a Euclidean line in is the image of an Euclidean line in under the canonical quotient map