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# Teichmüller geometry: a primer

The genus g Teichmüller space Teich(g) is the space of all marked hyperbolic metrics on a genus g surface $\Sigma_g$, modulo homotopy, or equivalently the space of all conformal structures, modulo biholomorphisms isotopic to the identity.

Recall it can be given a topology in a number of ways. Could we also give the space a geometry?

### Infinitesimally speaking

In line with the general philosophy of deformation theory, we may attempt to get a handle on the local geometry of this space—if indeed it exists—by describing infinitesimal deformations of hyperbolic metrics or conformal structures.

How could we describe deformations? One way is to by using a pair of transverse measured foliations on our surface—roughly speaking, “nice” partitions of the surface (minus finitely many singular points) into 1-dimensional manifolds—which record the directions of maximal and minimal stretch of an deformation on the surface. These can be encoded, complex analytically, using holomorphic quadratic differentials.

Alternatively, and in some sense more directly, we may describe infinitesimal deformations using quasiconformal maps—continuous maps whose infinitesimal distortion is uniformly bounded, from both above and below—which, by the measurable Riemann mapping theorem (which despite its name is not directly related to the Riemann mapping theorem) may be encoded, complex analytically, using Beltrami differentials.

Now recall that tangent vectors are, in general, derivations, i.e. infinitesimal deformations, and cotangent vectors are what tangent vectors—which are canonically identifiable with cocotangent vectors—eat (for variety’s sake, they also eat Koko Krunch for breakfast.) With this in mind, we assert (without quite proving, although one could) that QD(S), the complex vector space of holomorphic quadratic differentials on S, is (or, rather, is isomorphic to, or in other words can be used to describe) the cotangent space to Teich(g) at [(S, h)], and BD(S), the complex vector space of Beltrami differentials modulo infinitesimally-trivial ones the tangent space.

### Distances and geodesics

This characterization of the tangent and cotangent spaces allows us to define a metric, the Teichmüller metric, by $d\left( [(S_1. h_1)], [(S_2, h_2)] \right) = \inf_f K_f$, where the infimum is taken over all quasiconformal homeomorphisms $f: S_1 \to S_2$, and $K_f$ is the dilatation of f.

By construction (and the properties of quasiconformal maps), this is symmetric and satisfies the triangle inequality; proving positivity requires Teichmüller’s existence and uniqueness theorems, which assert that there exists a unique (up to homotopy) minimal-dilatation quasiconformal map between any two points in Teichmüller space.

The Teichmüller metric is Finsler, not Riemannian: it is a $L^\infty$-type norm; more precisely, we may show (after some relatively straightforward, but non-trivial work) that the norm it induces on the tangent spaces is given by $\mu \mapsto \sup \mathrm{Re} \int \mu q$, where the supremum is taken over all holomorphic quadratic differentials q with $\|q\| = 1$.

(In particular, the norm is dual to a $L^1$-type norm on QD(S), is hence strictly convex, and in particular not something induced by a bilinear form—in other words, not Riemannian.)

With a metric in hand we can ask what the geodesics in this space look like, i.e. what are the “straight lines” (which are the geodesics in our familiar Euclidean space) in this space?

Well, Teichmüller geodesics (sometimes called Teichmüller lines) are given by fixing a holomorphic quadratic differential—or, from a geometric point of view, fixing a pair of transverse measured foliations on the surface—and then scaling that differential, i.e. continuously stretching and contracting along these foliations. What this does to the hyperbolic metric / conformal structure is fairly straightforward to visualize on a single coordinate patch, but globally, on the surface as a whole, the effect can be rather more complicated.

[One day I’ll have a visualization here. Maybe. Hopefully.]

This characterization of Teichmüller geodesics can also be used to show that the Teichmüller metric is (geodesically, and hence metrically by Hopf-Rinow) complete.

### Curvature

The Teichmüller metric is not negatively-curved. Since we are not dealing with a Riemannian metric here, we need to specify what we mean here, and what we will sketch below is that the metric is not negatively-curved in the sense of Busemann: distinct geodesic rays originating from the same point do not diverge exponentially.

Masur constructed more-or-less explicit examples of distinct geodesic rays (in his thesis, which then became an Annals paper) which stay a bounded distance apart. To build these, he starts with a pair of Strebel differentials—holomorphic quadratic differentials whose horizontal foliations consist of closed leaves, which may be grouped into finitely many annuli, which together with a finite number of singular leaves form a partition of the surface—with the same annuli but different relative lengths between the annuli. (Ralph Strebel proved that such differentials always exist, with prescribed relative lengths, hence their name.)

Now do affine stretches along each annuli to obtain the geodesic rays, which are called Strebel rays. The construction of the Strebel differentials, and more precisely the structure of their horizontal foliations, ensures that corresponding points on the resulting rays are always bounded quasiconformal dilatation, i.e. bounded Teichmüller distance, apart.

### Symmetries

We may also ask: what are the isometries of this space? Or, slightly less precisely but perhaps more descriptively, what symmetries does this space have?

To answer this, we define the mapping class group Mod(g) (sometimes called the Teichmüller  modular group) as the group of all orientation-preserving homeomorphisms $\Sigma_g \to \Sigma_g$, modulo those which are homotopic to the identity.

It is clear that any element of the mapping class group induces an isometry of Teichmüller space: pre-composing a Teichmüller map with any representative of mapping class still yields a Teichmüller map.

It is a result of Royden that the converse is also true: any isometry of Teichmüller space (modulo the hyperelliptic involution in the case of genus 2) comes from the mapping class group.

The key lemma involved, which Royden proves by mucking around with the space of holomorphic quadratic differentials and its analytic / algebraic underpinnings, is this: any complex linear isometry between the (co)tangent spaces induces a conformal map between the underlying points in Teichmüller space—which, recall, may be thought of as Riemann surfaces.

(In fact Royden’s lemma—really a theorem—characterizes these linear isometries very precisely: any such complex linear isometry $\varphi$ is given by $\varphi(\eta) = \alpha\eta \circ \psi$, where $\alpha \in \mathbb{C}$ has modulus 1 and $\psi$ is a conformal map between the surfaces, although we will not need the full strength of it here.)

Now any isometry induces complex linear isometries between cotangent spaces and between tangent spaces, and hence conformal maps—in particular orientation-preserving self-homeomorphisms, which are well-defined mod homotopy, or in other words, mapping classes—between the underlying points.

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# Isomorphic groups via hyperbolic geometry

$\mathrm{PSO}(1, 2)$ is the isometry group of hyperbolic 2-space under the hyperboloid model; $\mathrm{PSL}(2,\mathbb{R})$ is the isometry group of the upper half-plane; $\mathrm{PSU}(1,1)$ is the isometry group of the Poincaré unit disk. Since these are in fact models for the same space, all of these Lie groups are isomorphic.

Question: is there a proof of this isomorphism that doesn’t proceed through this identification as isometry groups? (Probably, via the Lie algebras for instance, but I’m a little too lazy to try and work it out at the moment.)

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# Mostow Rigidity: several proofs

Mostow rigidity (or, technically, the special case thereof, for $\mathrm{PO}(1,3)$) states that if M and N are two closed (i.e. compact and boundary-less) hyperbolic 3-manifolds, and $f: M \to N$ is a homotopy equivalence, then f is homotopic to an isometry $g: M \to N$.

This is a strong rigidity result. If M and N were hyperbolic surfaces, by contrast, there are plenty of homotopy equivalences between them which are not homotopic to isometries; indeed, this is the very subject of Teichmüller theory (technically, we’re looking at moduli space, not Teichmüller space, but eh.)

We can state it in algebraic terms, by identifying the hyperbolic 3-manifolds $M \cong \mathbb{H}^3 / \Gamma_M$ and $N \cong \mathbb{H}^3 / \Gamma_N$ with their respective fundamental groups $\Gamma_M = \pi_1(M)$ and $\Gamma_N = \pi_1(N)$, which are (uniform) lattices in $\mathrm{Isom}^+(\mathbb{H}^3) \cong \mathrm{PO}(1,3)$.

Mostow rigidity then states any isomorphism between the lattices is given by conjugation in $\mathrm{PO}(1,3)$. Again, contrast this with the case of lattices in $\mathrm{Isom}^+(\mathbb{H}^2) \cong \mathrm{PO}(1,2)$ (i.e. fundamental groups of hyperbolic surfaces), which have plenty of outer automorphisms (elements of the [extended] mapping class group.)

Here we describe, at a relatively high level, various proofs of this result.

### Step 1: lift and extend

Let $\tilde{h}: \mathbb{H}^3 \to \mathbb{H}^3$ be a lift of h to the universal cover(s). This map is a quasi-isometry, by a Milnor-Švarc argument:

• The Cayley graphs of $\Gamma_M$ and $\Gamma_N$ are both quasi-isometric to $\mathbb{H}^3$ by Milnor-Švarc, the quasi-isometry from the Cayley graphs into hyperbolic space being the orbit map.
• If we now construct quasi-inverses from hyperbolic space to the Cayley graphs, then $\tilde{h}$ is coarsely equivalent to a map between the Cayley graphs obtained by composing one orbit map with the quasi-inverse of the other. To wit: $\tilde{h}$ sends $\gamma x_0$ to $(h_* \gamma) x_0$, and the induced map $h_*$ is an isomorphism of the fundamental groups

The quasi-isometry $\tilde{h}$ from the hyperbolic space $\mathbb{H}^3$ to itself extends to a self-homeomorphism of the Gromov boundary $\partial \mathbb{H}^3$:

This is a standard construction which holds for any quasi-isometry between hyperbolic spaces in the sense of Gromov: the boundary extension is defined by $\partial\tilde{h}([\gamma]) = [\tilde{h} \circ \gamma]$, and Gromov hyperbolicity is used to show that this map is a well-defined homeomorphism; roughly speaking, the idea is that quasi-isometries are like “biLipschitz maps for far-sighted people”, and at the boundary at infinity, the uniformly bounded distortions become all but invisible.

### Step 2: boundary map is quasiconformal

Now we claim that $\partial\tilde{h}: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is quasiconformal.

This follows e.g. from the following geometric lemma: if L is a geodesic and P a totally-geodesic hyperbolic plane in $\mathbb{H}^3$ with $L \perp P$, then $\mathrm{diam}(\pi_{\hat(L)}(\tilde{h}(P)) \leq D$ for some constant D depending only on the quasi-isometry constants of $\tilde{h}$. (Here $\hat{L}$ is the unique geodesic uniformly close to the quasigeodesic $\tilde{h}(L)$.)

The proof of the lemma uses three key ingredients from the geometry of hyperbolic space: thinness of ideal triangles, quasigeodesic stability—which says that the image of any geodesic segment by a quasi-isometry is uniformly (depending only on the quasi-isometry constants) close to the actual geodesic with the same endpoints, which we will call the “straightening” of the quasigeodesic—, and the (1-)Lipschitz property of projection to a geodesic.

Let $x \in L \cap P$ be the unique point of intersection. If $y \in P$, let z be the point in $\partial_\infty P$ corresponding to the ray from x through y. Now consider the ideal triangle with L as one of its sides and as the third vertex. h(x) is uniformly close to the images of the two sides ending at z , and also to straightenings $\bar{M_1},\bar{ M_2}$ thereof; since projection to a geodesic is Lipschitz, the projection of h(x) to the straightening $\hat{L}$ of the quasigeodesic h(L) is uniformly close to the projections of $\bar{M_1}, \bar{M_2}$.

But now the image of the geodesic ray [x, z] lies uniformly close to some geodesic ray, whose projection onto $\bar{L}$ lies between the projections of h(x) and h(z). Hence any point on this geodesic ray has image which projects between h(x) and h(z); again using the Lipschitz property of the projection, we obtain uniform bounds on the diameter we are trying to control, as desired.

Now note that any circle in $\partial\mathbb{H}^3 \cong \hat{\mathbb{C}}$ is the boundary of a totally geodesic hyperbolic plane in $\mathbb{H}^3$, and our lemma then tells us that the image of the circle under $\tilde{h}$ has its outradius/inradius ratio bounded above by $e^D$.

### Step 3: boundary map is conformal

And then we would be done, for this implies $\tilde{h}$ is (up to homotopy) a hyperbolic isometry, which descends to an isometry $h: M \to N$.

#### An argument along the lines of Mostow’s original

Suppose $\tilde{h}$ is not conformal. Consider the $\Gamma_M$-invariant measurable line field $\ell_h$ on $\hat{\mathbb{C}}$ obtained by taking the direction of maximal stretch at each point.

Rotating this line field by any fixed angle $\phi$ yields a new measurable line field $\ell_h^\phi$. But now $\bigcup_{0 \leq \phi \leq \pi/2} \ell_h^\phi$ yields a measurable $\Gamma_M$-invariant proper subset of $T^1M$, which contradicts the ergodicity of the geodesic flow on $T^1 M$.

An argument following Tukia, in the spirit of Sullivan

Suppose $\tilde{h} =: \phi$ is differentiable at z. Let $\gamma$ be a Möbius transformation fixing z and $\infty$, with z an attracting fixed point and with no rotational component. Now, by the north-south dynamics that hyperbolic Möbius transformations exhibit, $\gamma^n \phi \gamma^{-n} \to d\phi_z$ as $n \to\ infty$; on the other hand, $\gamma^n \phi \gamma^{-n}$ conjugates $\gamma^n\Gamma_1\gamma^{-n}$ to $\gamma^n\Gamma_2\gamma^{-n}$.

Again, using convergence dynamics, we have $\gamma^n\Gamma_i\gamma^{-n} \to \Gamma_i^\infty$ as $n \to \infty$, for i = 1, 2, where both $\Gamma_i^\infty$ are cocompact and of equal covolume.

Now $d\phi_z$ conjugates $\Gamma_1^\infty$ to $\Gamma_2^\infty$; thus $\phi$ is conformal a.e., and hence conformal.

### Alternative perspective I: the Gromov norm

Gromov gives a proof which follows Step 1 as above, but then proceeds differently: he uses the Gromov norm to show that $\tilde{h}$ is conformal, without (separately) showing that it is quasiconformal. We sketch an outline of this here; see Calegari’s blog and/or Lücker’s thesis for details.

The Gromov norm is a measure of complexity for elements of the (singular) homology—more precisely, it is the infimum, over all chains representing the homology element, of $L^1$ norm of the chain.

It is a theorem of Gromov that the Gromov norm of (the fundamental class of) an oriented hyperbolic n-manifold M is equivalent, up to a normalization dependent only on n, to the hyperbolic volume of M. The proof proceeds by

• observing that it suffices to take the infimum over rational chains of geodesic simplices; this suffices to give the lower bound, with a constant given in terms of volume of an ideal simplex;
• using an explicit construction involving almost-ideal simplices with almost-equidistributed vertices to obtain a upper bound.

Then, since $\tilde{h}$ is a homotopy equivalence (and by looking at this last construction carefully), it—or, rather, its extension to $\overline{\mathbb{H}^3}$—sends regular simplices close to regular ideal simplices. Simplices in the chains from the upper bound construction above are (almost) equidistributed, and the error can be made arbitrarily small as we use the construction to get arbitrarily close to the bound.

Translating the vertices of our ideal simplices, we get a dense set of equilateral triangles that are sent by $\partial\tilde{h}$ to equilateral triangles on the boundary. This implies $\partial\tilde{h}$ is conformal a.e., and hence conformal.

(In short: Gromov’s theorem and its proof allows us to control volume in terms of explicitly-constructed singular chains and to control what happens to simplices in those chains under our homotopy equivalence.)

### Alternative perspective II: dynamically characterizing locally-symmetric Riemannian metrics

Besson, Courtois and Gallot have an entirely different proof, which uses a characterization of locally-symmetric Riemannian metrics in terms of entropy to conclude that, since f is a degree-1 map between locally-symmetric spaces, M and N are homothetic. It then follows from e.g. an equal-volume assumption that M and N would be isometric.

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# Arguments using moduli spaces: some examples

This blogpost started as a response to the question “why moduli spaces?” … but then David Ben-Zvi had a good pithy answer to that in his  Princeton Companion to Mathematics article on moduli spaces : “Moduli spaces can be thought of as geometric solutions to geometric classification problems.”

If the geometry of the moduli space is well-chosen, it can tell us things about the geometry of the spaces / objects which points in it correspond to. Ben-Zvi has a thoroughly concrete and persuasive discussion of this in the case of 1-dimensional real projective space, a.k.a. the moduli space of lines through the origin in the plane.

Below we document, somewhat briefly, a couple more examples of how we can say things properties of geometric objects of a certain type by arguing in a moduli space of all such objects.

### An example from (classical) algebraic geometry

A conic (or conic section) is a curve obtained by the intersection of a cone with a plane; familiar examples, known to the Greeks, include circles, ellipses, and hyperbolae.

Conics in 2-dimensional (complex) projective space $\mathbb{P}^2 = \mathbb{P}(\mathbb{C}^3)$ are parametrized by $\mathbb{P}(\mathrm{Sym}^2 k^3) \cong \mathbb{P}^5$; explicitly, we have the correspondence $[a:b:c:d:e] \leftrightarrow ax^2 + bxy + cy^2 + dyz + ez^2 = 0$.

In other words, $\mathbb{P}^5$ is the moduli space of conics in $\mathbb{P}^2$. We will use this to show that given any five distinct points $(p_1, p_2, p_3, p_4, p_5)$ in $\mathbb{P}^5$ such that no four lie on a (projective) line, there is a unique conic (possibly degenerate) passing through $(p_1, p_2, p_3, p_4, p_5)$.

The conics through any given point in $\mathbb{P}^2$ are parametrized by a hyperplane in the moduli space $\mathbb{P}^5$; conics through 5 distinct points are given by the intersection of five hyperplanes, which is non-empty by considerations of dimension. This already gives us existence, even without the “no four on a line” condition; it remains only to show uniqueness given this additional condition.

Suppose this intersection consists of a single point (in $\mathbb{P}^5$, i.e. a unique conic in $\mathbb{P}^2$.) Then it is clear that we cannot have any four of our $p_i$ on a line $\ell$, for taking the union of this line and a line between the remaining point and any one of the four yields a conic containing the five given points, contradicting uniqueness.

Conversely, suppose the intersection is larger, i.e. it contains a $\mathbb{P}^1$ (a “pencil”) of conics.

We parametrize this pencil by $\{sF + tG | [s:t] \in \mathbb{P}^1\}$ for some $F, G$. Then, for any $q \in \mathbb{P}^2$ on the line containing $p_i, p_j$, $sF(q) + tG(q) = 0$ has a solution (in $s, t$), i.e. there is a conic in our pencil going through $q$. We may verify that any conic in $\mathbb{P}^2$ containing three collinear points is a pair of lines (or a double line); by the pigeonhole principle, at least one of these lines $\ell$ must contain at least three of the $p_i$.

Now if $\ell$ contains exactly three of the $p_i$, then there is a unique conic containing all five $p_i$—the pair of lines consisting of $\ell$ and the line through the remaining two $p_i$. Since by assumption we do not have uniqueness, $\ell$ must contain (at least) four of the $p_i$.

### An example from geometric topology

The genus g Teichmüller space Teich(g) is the space of all complex / conformal structures on a closed Riemann surface of genus g, up to some natural notion of equivalence. Equivalently, via uniformization, it is a space of all constant-curvature Riemannian metrics on a topological surface of genus g.

We will focus on the case of surfaces of genus at least two; for these surfaces, uniformization tells us that the constant-curvature metrics are hyperbolic. There is an analogous theory for genus one which gives us the moduli space of flat metrics on the torus, and many of the same general arguments that appear below can be applied there as well, but the specifics there can have a slightly different flavour, due to the lack of negative curvature (in the objects, that is.)

A point in Teich(g) can be specified by a pair (S, h), where S is a surface with a hyperbolic metric, and $h: \pi_1(F) \to \pi_1(S)$ is a marking, a choice of isomorphism from the fundamental group of a “naked” topological (reference) surface F without a metric and our metrized surface S. To steal Dick Canary’s metaphor: S is the hyperbolic clothing, and h provides instructions for how to wear it.

Technically, a point in Teich(g) is an equivalence class of such pairs, where two marked hyperbolic metrics $(S_1, h_1)$ and $(S_2, h_2)$ are considered equivalent if there is an isometry $j: S_1 \to S_2$ such that $j \circ h_1 \simeq h_2$, i.e. the two maps are homotopic to each other. In terms of the metaphor: jiggling the clothing around a little doesn’t give an essentially different way of wearing it.

We have specified Teich(g) as a set. We can give it a topology in (at least) two ways:

1. By identifying each equivalence class of marked hyperbolic structures $[(S,h)]$ with a holonomy representation $h_*: \pi_1(F) \to \mathrm{Isom}^+(\mathbb{H}^2) \cong \mathrm{PSL}(2,\mathbb{R})$ and thus Teich(g) with a subspace of the representation variety $X_2(F) := \mathrm{Hom}(\pi_1(F), \mathrm{PSL}(2,\mathbb{R})) / \mathrm{PSL}(2,\mathbb{R})$, where the quotient identifies representations that are conjugate in $\mathrm{PSL}(2,\mathbb{R})$—this corresponds in the algebra to the identification of homotopic marked metrics in the geometry.$X_2(F)$ isn’t actually a variety, in the algebraic geometry sense, but it does have natural topology. The induced topology on Teich(g) is called the algebraic topology.
2. By defining a metric on Teich(g), the Teichmüller metric $d_{Teich}$, and giving it the metric topology. $d_{Teich}([(S_1, h_1)], [(S_2, h_2)])$ is defined as (the logarithm of) the minimum quasiconformal distortion over all quasiconformal maps between the two marked hyperbolic structures.(For full definitions see e.g. Chapter 11 of Farb and Margalit’s Primer on Mapping Class Groups.)

A moment’s thought and some fiddling shows that these two approaches produce the same topology.

Intuitively, neighborhoods in this topology correspond to sets of marked hyperbolic metrics which don’t differ from each other too much—the clothing has similar dimensions, and the instructions are relatively similar.

### A space to act on

The mapping class group Mod(F) of a surface (or more generally of any space) F is defined as the group of all isotopy classes of homeomorphisms from to itself. Mod(F) may be thought of as the group of symmetries of a surface. The notation Mod comes by way of analogy with SL(2, Z), which is the mapping class group of the torus—the mapping class group for a surface is sometimes also called the Teichmüller modular group.

The mapping class group of a genus g surface acts naturally on the genus g Teichmüller space by change of marking: $\varphi \cdot [(S,h)] = [(S, \varphi_* h)]$, i.e. by changing the layout of the naked topological surface, it yields effectively different instructions for how to wear the hyperbolic clothing.

It can be shown that this action is properly discontinuous (see e.g. section 12.3 of Farb and Margalit), and isometric if we give Teich(g) the Teichmüller metric. The quotient of Teich(g) is $\mathcal{M}_g$, the moduli space of genus g surfaces—which we discuss slightly more below,

Using this action, we may obtain Thurston’s classification of elements of Mod(S), analogous to classification of elements of SL(2, R) by their action on hyperbolic 2-space: define the translation length $\tau(\phi) := \inf_{\mu \in \mathrm{Teich}(g)} d_{Teich}(\mu, \phi(\mu))$; then

• periodic elements $\phi$ have fixed points in Teichmüller space, and hence have $\tau(\phi) = 0$ and achieve this inf (analogous to elliptic elements of SL(2, R));
• pseudo-Anosov elements $\phi$ have positive translation length, and always achieve this translation length on a Teichmüller geodesic (analogous to hyperbolic elements of SL(2, R));
• reducible elements $\phi$ have zero translation length, but do not achieve it (analogous to parabolics.)

A fuller description of this classification and the work needed to obtain it can be found in Chapter 14 of Farb and Margalit’s Primer.

### A space to explore

It would perhaps been possible to rephrase the above classification and obtain it without the use of Teichmüller space, but the use of Teichmüller space certainly helps illuminate the (geometric) structure of the argument and how it is analogous to the classification of elements of SL(2, R) essentially using hyperbolic geometry.

Similarly, moduli spaces can provide a new light in which to consider / describe possible geometric structures.

For instance, we might ask: how could we imagine or describe the various hyperbolic metrics can we put on a genus g surface?

One possible approach to answering this could involve putting coordinates on the genus g Teichmüller space and attempting to interpret those coordinates in terms of the genus 2 surface. Indeed, we can put a set of (global!) 6– 6 coordinates on Teichmüller space, the Fenchel-Nielsen coordinates, which we obtain by considering a set of 3g – 3 disjoint geodesics on our surface (a pants decomposition, so called because it divides our surface into spheres with three boundary components, i.e.  pairs of pants) and taking length and twist parameters along those geodesics—for a full description see Danny Calegari’s post (linked right above).

The Fenchel-Nielsen coordinates (together with a little hyperbolic geometry) give us a very concrete way of answering our question: to obtain some hyperbolic metric on a genus 2 surface, say, we

1. take two pairs of pants,
2. identify pairs of boundary curves that we will glue to obtain our genus 2 surface,
3. specify how long we want each pair of boundary curves to be—this uniquely determines the hyperbolic metric on each pair of pants—,
4. specify how much we twist the curves in each pair relative to each other when we glue them together.

To obtain a different metric, we change our specifications for the lengths and/or the twist parameters.

Or, jumping up a dimension, we might ask: are there any hyperbolic metrics on closed 3-manifolds that we could describe similarly concretely?

One way to answer this would be to consider a corresponding representation variety $\mathrm{Hom}(\pi_1(M), \mathrm{PSL}(2,\mathbb{C})) // \mathrm{PSL}(2,\mathbb{C})$. Unfortunately, 3-manifold fundamental groups are much more complicated than surface groups, and these representation varieties have been much harder to study (although, using machinery I do not [yet?] understand, they are actually algebraic varieties.)

### Deformation, degeneration, and the totality

Putting the totality of geometric objects together as a moduli space highlights ways of varying the geometry on them which may be less obvious or natural from a non-moduli-space perspective:

• Infinitesimal deformations of a particular geometric structure: e.g. we might ask, how many degrees of freedom are there if we start with a given hyperbolic metric and try to vary the lengths of the geodesics? Can we concretely describe or characterize these degrees of freedom? Barry Mazur has an excellent overview article which talks more about deformations.
• Degeneration of particular aspect/s of our geometric structures—e.g. what happens to our hyperbolic metric if we pick a geodesic loop and shrink its length towards zero, or expand it towards infinity?

Having such a structured totality also allows us to answer with some degree of precision questions such as “how special is such a feature (say, the presence of many short disjoint geodesics) for this type of geometric structure?” With a moduli space in hand, we might be able to answer “it is a high-codimension feature” (so rather special), or perhaps “it is true except in small balls of finite total volume” (so not very special.)

### Aside: coarse and fine moduli; universal properties

The distinction between Teichmüller space Teich(g) and its quotient moduli space $\mathcal{M}_g$, briefly mentioned above, more generally reflects a distinction between fine and coarse moduli spaces. The emphasis on marked hyperbolic metrics may not seem all that natural: we may not care so much about different instructions for how to wear the clothing, but are only interested in essentially different sorts of hyperbolic clothing—which is, untranslating the metaphor, what $\mathcal{M}_g$ is describing.

$\mathcal{M}_g$, though, is not as nice as Teich(g)—for one thing, Teich(g) has the universal property that any continuously-varying family of marked genus g surfaces parametrized by a topological space S is described by a continuous map from S into Teich(g). Indeed, we can put a complex structure on Teich(g), and then we can replace “continuous” above with “complex analytic”.

$\mathcal{M}_g$, however, does not have this property, essentially because any higher-genus surface has non-trivial automorphisms, which kill any hope of obtaining the universal property; to address this problem, we “rigidify” the surface (or, from another perspective, “kill off the automorphisms”) by specifying a marking.

Such a universal property provides one way of specifying, categorically, what a moduli space is, and is one way of more precisely expressing the notion that the moduli space captures the geometry that we are interested in.

For more on this circle of ideas—really quite key to the history of Teichmüller space—, introduced by Teichmüller in his 1944 paper and expanded upon by Grothendieck in a series of lectures at the Séminaire Henri Cartan, see this article of A’Campo, Ji, and Παπαδόπουλος.

### Invariants from moduli space

Moduli spaces also allow for a novel way of defining topological invariants, which, again, Ben-Zvi’s article describes much better than I could:

‘an important application of moduli spaces in geometry and topology is inspired by quantum field theory, where a particle, rather than follow the “best” classical path between two points, follows all paths with varying probabilities. Classically, one calculates many topological invariants by picking a geometric structure (such as a metric) on a space, calculating some quantity using this structure, and finally proving that the result of the calculation did not depend on the structure we chose. The new alternative is to look at all such geometric structures, and integrate some quantity over the space of all choices. The result, if we can show convergence, will manifestly not depend on any choices. String theory has given rise to many important applications of this idea, in particular by giving a rich structure to the collection of integrals obtained in this way. Donaldson and Seiberg-Witten theories use this philosophy to give topological invariants of four-manifolds. Gromov-Witten theory applies it to the topology of symplectic manifolds, and to counting problems in algebraic geometry.’

We remark here that Vassiliev invariants in knot theory can be seen as another example of this approach; this is perhaps clearer from the point of view of the Kontsevich integral, rather than from the combinatorial perspective.

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# Ergodicity of the Geodesic Flow

### The geodesic flow

Given any Riemannian manifold M, we may define a geodesic flow $\varphi_t$ on the unit tangent bundle $T^1M$ which sends a point (x, v) to the point $(\varphi_t x, \varphi_t^* v)$, where

• $\varphi_t x$ is the point distance from x along the geodesic ray emanating from x in the direction of v, and
• $\varphi_t^* v$ is the parallel transport of v along the same ray

(it’s a mouthful, isn’t it? It’s really simpler than all those words make it seem.) Note, at each point, we remember not just where we are—the point $x \in M$—, but also where we’re going—the direction vector $v \in T_x M$; if we were to forget this second piece of information, we would become a little unmoored: here we are … where should we go next?

### Ergodicity

When M is a closed (i.e. compact, no boundary) hyperbolic surface, or more generally closed with strictly negative curvature, this geodesic flow is ergodic, i.e. any subset of $\Sigma$ or M invariant under the flow has either zero measure, or full measure. Here the measure on our Riemannian manifold is the pushforward of the Lebesgue measure through the coordinate charts.

Since linear combinations of step functions are dense in the space of bounded measurable functions, we may equivalently define ergodicity as: any measurable function invariant under the flow is a.e. constant.

(Side note: with more assumptions on the curvature we may relax the compactness assumption to a finite volume assumption)

### The Hopf argument (for closed hyperbolic manifolds)

This is essentially due to the exponential divergence of geodesics in negative curvature , and the splitting of the tangent spaces $T_ v T^1M = E^s_v \oplus E^0_v \oplus E^u_v$ into stable, tangent (flowline), and unstable distributions; these give rise to three maximally transverse foliations, the stable foliation $W^s$, the unstable foliation $W^u$, and the foliation by flowlines $W^0$.

The flow is exponentially contracting in the forward time direction on the leaves of the stable foliation $W^s$, and on which the flow is exponentially contracting in the reverse time direction the leaves of the unstable foliation $W^u$. In other words, the flow is Anosov.

We may describe these foliations explicitly in the case of constant negative curvature—if we take $\gamma$ to be the geodesic tangent to $v \in T^1M$,

• $W^s$(v) is (the quotient image of) the unit normal bundle to the horosphere through $\pi(v) \in M$ tangent to the forward endpoint of $\gamma$ in $\partial_\infty\mathbb{H}^n \cong \partial_\infty\widetilde{M}$. “forward” here being taken with reference to how v is pointing along $\gamma$;
• $W^u(v)$ is (the quotient image of) the unit normal bundle to the horosphere through $\pi(v)$ tangent to the backward endpoint of $\gamma$ in $\partial_\infty\mathbb{H}^n \cong \partial_\infty\widetilde{M}$;
• $W^0(v) = \gamma$.

#### Step 1

Suppose f is a $\phi$-invariant function; by replacing f with min(f, C) if needed, WMA f is bounded. Since continuous functions are dense in the set of measurable functions on M, we may approximate f in $L^1$ by bounded continuous functions $h_\epsilon$.

By the Birkhoff ergodic theorem, forward time averages [w.r.t. $\phi$] exist for $h_\epsilon$.

By an argument involving the $\phi$-invariance of f and the triangle inequality, f is well-approximated (in $L^1$) by the forward time averages of $h_\epsilon$.

#### Step 2

The forward time averages of $h_\epsilon$ are constant a.e., since by invariance these averages are already constant a.e. on (each of) the leaves of $W^0$, and they are also constant a.e. on (each of the) unstable and stable leaves, by uniform continuity of $h_\epsilon$.

#### Step 3

To conclude that time averages, and hence our original arbitrary integrable function, are constant a.e. on M, we (would like to!) use Fubini’s theorem: locally near each $(x_0,v_0) \in T^1M$, the set of (x, v) along each of the foliation directions at which the time averages are equal to those at $(x_0,v_0)$ has full measure, by the previous Step.

By Fubini’s theorem applied to the three foliation directions, we (would) conclude that the set of nearby (x, v) at which the time averages are equal to those at $(x_0,v_0)$ has full measure. Hence the time averages are locally constant, and since $T^1M$ is connected we are done.

### But! (Also more generally, for K < 0)

The problem is that while our stable and unstable leaves are differentiable, the foliations need not be—i.e. the leaves may not vary smoothly in their parameter space.

To justify the use of a Fubini-type argument one instead shows that that these foliations are absolutely continuous.

The proof then immediately generalizes to all compact manifolds with (not necessarily constant) negative sectional curvature. For more general negatively-curved nanifolds, the stable and unstable foliations $W^s$ and $W^u$ may still be described in terms of unit normal bundles over horospheres, where horospheres are now described, more generally, as level sets of Busemann functions.

The proof of absolute continuity of the foliations proceeds as follows

1. Showing that the stable and unstable distributions $E^s$ and $E^u$ (also the “central un/stable” or “weak un/stable” distributions, i.e. $E^{s0} := E^s \oplus E^0$ and  $E^{u0} := E^u \oplus E^0$) (of any $C^2$ Anosov flow) are Hölder continuous—i.e. given $x, y \in M$, the Hausdorff distance in $TTM$ between the stable subspace $E^s(x)$ and the stable subspace $E^s(y)$ is $\leq A \cdot d(x,y)^\alpha$.
Roughly speaking, this is true because any complementary subspace to $E^s$ will become exponentially close to $E^s$ under the repeated action of the geodesic flow, by the same mechanism that makes power iteration tick; and the distance function on M is Lipschitz. Analyzing the situation more carefully, and applying a bunch of simplifying tricks such as the adjusted metric described in Brin’s section 4.3, yields the desired Hölder continuity.
2. Using this, together with the description of horospheres as limits of sequences of spheres with radii increasing to $+\infty$, to establish that between any pair of transversals for the un/stable foliation, we have a homeomorphism which is $C^1$ with bounded Jacobians, and hence absolutely continuous.
Very slightly less vaguely, Hölder continuity of $E^{u0}$, together with the power iteration argument as above, implies tangents to transversals to the stable foliation $W^s$ become exponentially close; given regularity of the Riemannian metric, this implies the Jacobians of the iterated geodesic flow on these transversals become exponentially close. By a chain rule argument and another application of the power iteration argument, this implies that the Jacobians of the map between transversals are bounded.
This condition on the foliations is known as transversal absolute continuity, and implies, by a general measure theoretic argument (see section 3 of Brin’s article), absolute continuity of the foliations.
3. Note that this last step, at least as presented in Brin, appears to require the use of pinched negative curvature.

### References

Eberhard Hopf, “Ergodic theory and the geodesic flow on surfaces of constant negative curvature.” Bull. Amer. Math. Soc. 77 (1971), no. 6, 863–877.

Yves Coudene, “The Hopf argument.

Misha Brin, “Ergodicity of the Geodesic Flow.” Appendix to Werner Ballman’s Lectures on Spaces of Nonpositive Curvature.

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Snippets

# The many faces of the hyperbolic plane

$\mathbb{H}^2$ is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

1. It is diffeomorphic to $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ as a topological space (or, indeed, isometric as a Riemannian manifold, if we give $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ a left-invariant metric): to show this, we note that $\mathbb{H}^2$ has isometry group $\mathrm{SL}(2,\mathbb{R}) / \pm\mathrm{id}$, and the subgroup of isometries which stabilize any given $p \in \mathbb{H}^2$ is isomorphic to $\mathrm{SO}(2)$.
2. Since any positive-definite binary quadratic form is given by a symmetric 2-by-2 matrix with positive eigenvalues, and since the group of linear transformations on $\mathbb{R}^2$ preserving any such given form is isomorphic to $\mathrm{O}(2)$, $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the space of positive-definite binary quadratic forms of determinant 1, via the map from $\mathrm{SL}(2,\mathbb{R})$ to the symmetric positive-definite 2-by-2 matrices given by $g \mapsto g^T g$.
3. $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}^2$ is also the moduli space of marked Riemann surfaces of genus 1, i.e. the Teichmüller space Teich(S) of the torus S. One way to prove this is to note that any such marked surface is the quotient of $\mathbb{R}^2$ by a $\mathbb{Z}^2$ action; after a suitable conformal transformation, we may assume that the generators of this $\mathbb{Z}^2$ act as $z \mapsto z + 1$ and $z \mapsto z + \tau$ for some $z \in \mathbb{H}^2$ (in the upper half-plane model.) But now $\tau$ is the unique invariant specifying this point in our moduli space.
4. Since any marked Riemann surface of genus 1 has a unique flat metric (inherited as a quotient manifold of the Euclidean plane), $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the moduli space of marked flat 2-tori of unit area.
5. Since there is a unique unit-covolume marked lattice associated to each marked complex torus in the above, $\mathrm{SL}(2,\mathbb{R}) / \mathrm{SO}(2)$ is also the space of marked lattices in $\mathbb{R}^2$ with unit covolume.
6. Note we may go directly between marked Riemann surfaces and quadratic forms by considering intersection forms.
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Articles

# Fuchsian vs. Kleinian

Or, all the fun you could have in one additional dimension.

The former, of course, may be viewed as a special case of the latter; the more general Kleinian case shares many, but not all, of the characteristics of the simpler Fuchsian case. In particular, there is still a broad three-way classification of elements as elliptic, parabolic, or hyperbolic / loxodromic, and many arguments still proceed by examining these elements and their fixed loci.

The additional dimension introduces some combinatorial / geometric complications into these arguments; in particular, it is sometimes easier and more enlightening in the three-dimensional case to work purely at the boundary $\partial\mathbb{H}^3 \cong S^2$ rather than with the whole space $\mathbb{H}^3$ when making fixed-point (or other) arguments.

Here we collect a few facts about Fuchsian / Kleinian groups, mostly in order to highlight some things that stay the same, and other things that are new, when we go up from 2 to 3 dimensions in this case.

In the below, n is 2 in the Fuchsian case, and 3 in the Kleinian case.

### Elementary (sub)groups

Elementary Kleinian groups are groups which have a finite orbit in $\partial\mathbb{H}^n$.. Any elementary group has a finite orbit in the boundary of size at most 2 (see result on the limit set below.)

Every subgroup of $\mathrm{PSL}(2,\mathbb{R})$ (and hence, a fortiori, any Fuchsian group) consisting of only elliptic elements (besides the identity) has a common fixed point, and hence is elementary (indeed, cyclic.)

More generally, every elementary Fuchsian group is cyclic, or conjugate to a group generated by $g(z) = kz$ and $h(z) =- -1/z$ (i.e. a hyperbolic and a elliptic switching the two ends of the axes.)

(Proof: Katok, Theorems 2.4.1 and 2.4.3)

Kleinian elementary groups include

• the dihedral, tetrahedral, octahedral, and icosahedral groups;
• the finite extension of a rank one or two free abelian group of parabolics by elliptics; there are finitely many possiblities for the orders of the extension, by ___;
• a cyclic loxodromic group, possibly extended by the elliptic with the same axis, and possibly extended again by an order-two elliptic exchanging the endpoints of the axis.

(Proof: Marden, section 2.3; attributed to Ford 1929.)

### Limit sets

The limit set $\Lambda(\Gamma)$ of a Fuchsian / Kleinian group $\Gamma$ is the set of limit points of $\Gamma$-orbits of $\mathbb{H}^n$.

Note, since $\Gamma \curvearrowright \mathbb{H}^n$ properly discontinuously, we have $\Lambda(\Gamma) \subset \partial\mathbb{H}^n$.

Proposition: $\Lambda(\Gamma)$ is the smallest $\Gamma$-invariant subset of $\partial\mathbb{H}^n$.

Proof: It is clear that $\Lambda(\Gamma)$ is $\Gamma$-invariant; conversely, $\Gamma$-invariant subset of $\partial\mathbb{H}^n$ must contain all of the limit points of the orbits of $\Gamma$, i.e. must contain all of $\Lambda(\Gamma)$.

If $|\Lambda(\Gamma)| \leq 2$, then $\Gamma$ is elementary.

Proposition: If $|\Lambda(\Gamma)| > 2$ then $\Gamma$ is either all of $\partial\mathbb{H}^n$, or is a perfect (hence uncountable) nowhere dense subset of the boundary

(Proof: Katok, Theorem 3.4.6 and Marden, Lemma 2.4.1)

Proposition: $\Lambda(\Gamma)$ is the closure of the set of hyperbolic / loxodromic fixed points

(Proof: Katok, Theorem 3.4.4; Marden, Lemma 2.4.1)

### Domains of discontinuity

The domain of discontinuity $\Omega(\Gamma)$ is $\partial\mathbb{H}^n \setminus \Lambda(\Gamma)$. It is the largest open subset of the boundary on which $\Gamma$ acts properly discontinuously.

Proposition: For a finitely-generated Kleinian group, if $\Omega(\Gamma)$ is non-empty, then $\Omega(\Gamma)$ has one, two, or infinitely many components, each of which is either simply or infinitely connected. There are additional rigidity results on $\Omega(\Gamma)$ if $\Gamma$ preserves one or more of the components.
(Marden, Lemma 2.4.2)

Remark: in the Fuchsian case with $n=2$, each component of $\Omega(\Gamma)$ is (trivially) an open interval, and hence simply-connected, and the result on the number of components is a straightforward corollary of the Proposition above characterizing the structure of $\Lambda(\Gamma)$ when it has cardinality larger than 2.

Proof: Uses Ahlfors finiteness in the general case (see e.g. Marden, Lemma 2.4.2 and Chapter 3.)

Contrasted with the relatively elementary arguments above, this is a rather deeper argument.

### Fundamental regions and geometric finiteness

A Fuchsian / Kleinian group has Dirichlet fundamental regions / polyhedra $D_p(\Gamma)$ determined by the intersection of the hyperbolic half-spaces $H_p(T) =\{z \in \mathbb{H}^n : d(z,p) \leq d(z,T(p)) \}$.

This is connected, convex, and locally-finite polygon / polyhedron—but not necessarily finite-sided. For Fuchsian groups, we have the following

Proposition: A Fuchsian group $\Gamma$ is geometrically finite iff it is finitely-generated

(Proof: Forward direction–Katok, Theorem 3.5.4; reverse direction–Katok, Theorem 4.6.1)

Siegel’s Theorem: Any Fuchsian group of finite covolume is geometrically finite

The analogous results for general Kleinian groups are more subtle:

Theorem (Marden): A Kleinian group $\Gamma$ is geometrically finite iff it is cocompact, except possibly for a finite number of rank one and rank two cusps, where the rank one cusps correspond to pairs of punctures on the boundary of $\mathbb{H}^3 / \Gamma$.

(The rank of a cusp is the smallest number of generators of the parabolic subgroup of $\Gamma$ fixing the corresponding parabolic fixed point.)

Lemma (Wielenberg) Any Kleinian group of finite covolume is geometrically finite.

Proof: uses the thick-thin decomposition, and Marden’s theorem above.

There is also the Ford fundamental region / polyhedron, defined in terms of isometric circles / hemispheres (in $\mathbb{H}^2$ or $\mathbb{H}^3$, resp.): given a Fuchsian / Kleinian group $\Gamma$, the Ford fundamental region $F(\Gamma)$ is the closure of the set of points in $\mathbb{H}^n$ exterior to all isometric circles / hemispheres of elements of $\Gamma$.

For $\Gamma$ Fuchsian, $F(\Gamma) = D_0(\Gamma)$ (Katok, Theorem 3.3.5.)

In the more general Kleinian case, $F(\Gamma)$ is a limit of Dirichlet fundamental polyhedra $D_p(\Gamma)$ as $p \to \partial\mathbb{H}^3$. (Marden, Lemma 3.5.2.)

### Offline references

• Svetlana Katok, Fuchsian Groups
• Albert Marden, Outer Circles

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