Geometric group theory (I)

Geometric group theory is, really, not so much a single coherent subfield as a somewhat disparate collection of things united by a single underlying philosophy—that the study of group actions on spaces can yield information about the groups (and also about the spaces, although that is perhaps less the remit of geometric group theory than of other subfield/s.)

This is a a quite natural way of looking at groups if we recall that historically, groups were first formulated as abstractions to capture the algebraic structures of sets of automorphisms—symmetry groups, Galois groups, matrix groups, monodromy groups, and the like.

A close cousin of this philosophy—or perhaps another way of articulating it—is the idea that groups themselves can be viewed as geometric objects (see below), and that this perspective can yield fresh insights on their algebraic and algorithmic properties.

To make these philosophies effective we need some assumptions on the group G, the space X, and the group action G \curvearrowright X. The following are by no means the only ones that can be made to work, but are the most common:

  • To get a sensible notion of geometry, we assume that  X is a geodesic metric space, i.e. X is equipped with a metric d, and between any two points x, y \in X there is some rectifiable path whose length is equal to d(x,y).
    (Note that we do not assume X is Riemannian or Finsler; in general we define the length of a path \gamma as \sup \sum d(x_i, x_{i+1}), where the sup is taken over all finite subdivisions of \gamma.)
    Sometimes this can be weakened to just assuming that X is a length space, i.e. we do not necessarily have paths which achieve d(x,y), but d(x,y) is still the infimum of lengths of rectifiable paths between x and y.
  • To obtain a notion of groups as geometric objects themselves, a word metric is usually used, and in order to define this G is assumed to be finitely-generated
  • In order for the group action to play nicely with the geometry, assumptions are usually made that are sometimes grouped under the umbrella term of “geometric action”: this almost always involves the action being by isometries (so that it preserves the geometry) and properly discontinuous (so that the quotient is still Hausdorff), and often but not always the action being co-compact.
    Below all of our group actions will be isometric and properly discontinuous.


The definition of a word metric starts with the choice of a generating set. This is a somewhat arbitrary choice, and thus the resulting metric is not quite an intrinsic object—a rather unsatisfying state of affairs. We would, after all, like to say things about the geometry of a group and try to relate those things to the algebraic and other properties of the group, rather than a specific presentation of the group.

To remedy this, we would like to declare that word metrics for the same group coming from different choices of generating sets are really equivalent. As it turns out, we can reach this identification geometrically by squinting a little, or slightly more precisely by taking coarse bi-Lipschitz equivalence, or, even more precisely, by declaring two metrics d_1, d_2 on \Gamma to be equivalent if there is a quasi-isometry f:(\Gamma, d_1) \to (\Gamma, d_2), i.e. a map satisfying \frac 1k d_1(x,y) - c \leq d_2(f(x), f(y)) \leq k d_1(x,y)+c for all x, y \in \Gamma.

One can show that quasi-isometry is an equivalence relation—the identity map is a quasi-isometry; compositions of quasi-isometries are quasi-isometries; there is a notion of a [coarse] quasi-inverse—and, then, that choosing a different generating set produces a equivalent (quasi-isometric) word metric.

The most common way of proving that two spaces are quasi-isometric is to produce an explicit quasi-isometry.

Using the orbit map as the explicit quasi-isometry (this takes some argument, but most of it can be fruitfully sketched in a diagram) leads to the Milnor-Švarc lemma, which states that any group \Gamma acting co-compactly on a geodesic metric space X, is quasi-isometric to (in the sense that some—and hence any—Cayley graph of \Gamma with the associated word metric is quasi-isometric to X.)

As a corollary, we may deduce that any two groups which act cocompactly on the same geodesic metric space are quasi-isometric to each other.

Quasi-isometric rigidity

Two relatively simple applications of the Milnor-Švarc lemma yield that finite subgroups and quotients of finitely-generated groups G are always quasi-isometric to G itself: for the former case we use the induced action of the subgroup on G; in the latter we case we consider the action of G on the quotient group. Thus commensurable finitely-generated groups are always quasi-isometric.

(This is also why geometric group theory is not the tool of choice for finite group theory—from this geometric point of view, finite groups are virtually trivial.)

Quasi-isometric groups are not always commensurable, but to produce (and prove) examples of this actually takes some work. Large families of familiar groups exhibit quasi-isometric rigidity, a phenomenon which may be loosely described as “quasi-isometry implies commensurability”, or sometimes just slightly more carefully as “quasi-isometry implies commensurability to another group in the same class.”

For instance, all hyperbolic surface groups—i.e. fundamental groups of closed oriented surfaces of genus at least 2—are quasi-isometric by Milnor-Švarc; since given any two such surfaces we can find a surface covering both surfaces with finite covering degree, they are also all (abstractly) commensurable.

Similarly, free groups are quasi-isometrically rigid, as are free abelian groups, nilpotent groups (although the most straightforward proof of this seems to use a beautiful but huge hammer), and so on.

One relatively easily-described example of pairs of quasi-isometric groups which are not abstractly commensurable to each other comes from fundamental groups  \pi_1(T^1\Sigma) of unit tangent bundles of hyperbolic surfaces \Sigma, which are uniform lattices in \widetilde{\mathrm{PSL}_2\mathbb{R}}. These are quasi-isometric to central extensions \pi_1(\Sigma) \times \mathbb{Z}, but by the structure of \widetilde{\mathrm{PSL}_2\mathbb{R}} are not commensurable to these central extensions.

A broader class of examples comes from topological, geometric, dynamical or algebraic properties not preserved by quasi-isometry, such as solvability, residual finiteness, or property (T).

Quasi-isometric invariants

This last paragraph leads naturally to the question of what properties or invariants of a group, if any, are invariant under quasi-isometry.

These invariants are sought not just in order to produce examples of quasi-isometric rigidity, but also for the allied but broader reason that they point to the limits of the tools of geometric group theory. This is a toolkit which, in general, will only distinguish things up to quasi-isometry, and isn’t of much use in trying to tell apart two groups which are quasi-isometric to each other (see, e.g., the earlier comment about finite groups.)

There is a long list of such invariants and properties: they include both large-scale / coarse geometrical, topological and dynamical properties such as word-hyperbolicity (see below), the ends of a group, and amenability, as well as more quantitative features such as growth, isoperimetric, and divergence functions, and cohomological dimension.

Groups with geometry: the role of nonpositive curvature

Groups—or, technically, quasi-isometric equivalence classes of Cayley graphs with word metrics—may considered as geometric objects in their own right, an idea first systematically articulated by Gromov.

Here cocompact group actions play an important role, by allowing us to invoke Milnor-Švarc to go between any other space our group may naturally act on (e.g. hyperbolic space, or a metric tree) and its the Cayley graph.

The richest results have been obtained where we have some notion of nonpositive, or even better, negative, curvature.

Hyperbolic groups

The prototypical case is that of Gromov-hyperbolic (or word hyperbolic, or \delta-hyperbolic, or sometimes just hyperbolic) groups.

A geodesic metric space X is defined to be \delta-hyperbolic if geodesic triangles in are \delta-slim, i.e. any point on any side of the triangle is within \delta of the (union of the) other two sides. A finitely-generated group G is \delta-hyperbolic if it acts cocompactly on some \delta-hyperbolic space. We may verify that hyperbolicity—though not the hyperbolicity constant—is a quasi-isometry invariant, and then, by Milnor-Švarc, this is equivalent to any Cayley graph for G being Gromov-hyperbolic.

This simple metric geometry notion somehow captures the coarse essence of negative curvature, as we shall see briefly below.

A key property of \delta-hyperbolic spaces is quasigeodesic stability, sometimes referred to as the Morse Lemma: any quasigeodesic segment (the image of a geodesic segment under a quasi-isometry) is uniformly close, with constants depending only on the quasi-isometry (k,c) and the hyperbolicity constant \delta, to an actual geodesic with the same endpoints, and vice versa.

Algorithmic and finiteness properties

Using the \delta-hyperbolicity condition, the triangle inequality, and quasigeodesic stability, we may show that k-local geodesics (i.e. rectifiable paths whose restrictions to segments of length at most k are geodesic) are global geodesics for any k > 8\delta.

This in turn gives rise to Dehn presentations for \delta-hyperbolic groups, which allow us to solve the word problem for these groups in “linear” time (or more precisely, in linearly many steps in the length of the word, although each step may itself take non-constant polynomial time.)

Hyperbolic groups also have good algorithmic properties beyond having solvable word problem: for instance, it may be shown that conjugate elements in a hyperbolic group are either representable by short words, or are conjugate by short words, and hence hyperbolic groups have solvable conjugacy problem; they are also geodesically biautomatic.

Somewhat surprisingly, the existence of Dehn presentations may be used to characterize hyperbolic groups—the reverse implication proceeds via the linear isoperimetric inequality (LIP) condition.

Dehn presentations can also be used to show that hyperbolic groups have finitely many conjugacy classes of finite-order elements; indeed, using the existence of coarse barycenters, we may show that hyperbolic groups have finitely many conjugacy classes of finite-order subgroups.

Negative curvature, coarsified

Besides coarse barycenters,  \delta_hyperbolic spaces admit appropriately coarsified versions of many negative-curvature features such as coarse projections, coarsely well-defined midpoints, and so on.

Quasigeodesic stability also allows us to construct boundaries for \delta-hyperbolic spaces and groups, which are in the first instance sets of directions at infinity. These sets may be naturally endowed with topologies; the resulting contraptions are known as Gromov boundaries, and are analogous to / generalisations of the boundary sphere of hyperbolic space.

They can be used to provide analogous proofs for results as the Tits alternative for subgroups of \delta-hyperbolic groups (cf. the Tits alternative for subgroups of \mathrm{SL}_2\mathbb{R}): isometries of \delta-hyperbolic spaces can be classified as elliptic, parabolic, or hyperbolic, in terms of translation lengths; the presence of two hyperbolic elements with disjoint fixed point sets allows us to play ping-pong; otherwise we may argue to obtain a virtually nilpotent group.

CAT(0) groups

If Gromov hyperbolicity is the appropriate notion of coarse negative curvature, it is less clear what makes for a good notion of coarse nonpositive curvature. One approach—which indeed applies more generally in the form of CAT(k) geometry—involves comparing geodesic triangles to geodesic triangles in corresponding model spaces (Riemannian manifolds) of constant sectional curvature. Specifically, CAT(0) spaces are spaces in which geodesic triangles are no fatter than corresponding comparison triangles in Euclidean space, and a CAT(0) group is one which acts co-compactly on some CAT(0) space.

Comparison geometry has a long and fine tradition, and CAT(0) spaces share many features of nonpositively curved spaces. Unfortunately, unlike hyperbolicity, the CAT(0) condition is not a quasi-isometry invariant, and in this sense being CAT(0) is rather less intrinsic of a property for a group.

CAT(0) spaces do still have nice properties: they are uniquely geodesic and contractible, for instance. CAT(0) groups have solvable word problem, and there is a fair amount we can say about isometries of CAT(0) spaces, aided by tools such as nearest-point retraction onto convex subspaces and well-defined barycenters. (That we do not see the word “coarse” here is one indication that this is in some ways a slightly different flavour of geometry compared to the Gromov-hyperbolic kind.) Analogous to the classification of isometries for hyperbolic spaces, we may classify isometries of CAT(0) spaces as elliptic, hyperbolic, or parabolic, depending on translation length (and whether it is achieved.)

We may define visual boundaries and ideal boundaries (also known as Tits boundaries) for CAT(0) groups in much the same way as we defined  for hyperbolic groups. However, here they are no longer quasi-isometry invariants; the homeomorphism type of the boundary we end up with in general could depend on the particular presentation we start with. This is not at all ideal, and there are any number of ways to remedy the situation; roughly speaking, the divergence between the various boundaries increases with how much flatness / zero curvature we see in our CAT(0) space.

In general, the pathologies that appear in CAT(0) geometry as compared to hyperbolic geometry are attributable to flat subspaces: quasi-flats—i.e. the images of flat subspaces under quasi-isometries—can be rather wild (here’s an exercise which illustrates one aspect of this: [singly-infinite, unbounded] logarithmic spirals are quasi-geodesics in the Euclidean plane.)

Indeed, the Flat Plane Theorem states that any proper co-compact CAT(0) space X (“co-compact” here meaning “admitting a co-compact geometric action by some group G“) is \delta-hyperbolic iff it does not contain a flat plane.

There is, however a fair amount we can say about flat subspaces in CAT(0) spaces and how they are reflected in the structure of CAT(0) groups. Starting from the Alexandrov Lemma (in some sense, a particular combination of the CAT(0) inequality, the triangle inequality and the law of cosines), we may use angle conditions to find flat triangles, flat quadrilaterals, and flat strips with product decompositions. Using the product decomposition theorem and other properties of hyperbolic isometries, we may establish the Flat Torus Theorem, which states (roughly speaking, in one formulation) that if a free abelian group of rank n acts co-compactly on a CAT(0) space X, then we can find a flat n-torus in the quotient of X by that action.

Cubulated groups

What if our groups acted not just on any old CAT(0) space, but on very particular CAT(0) spaces, which had tractable combinatorial structures on them? What if, even better, these particular CAT(0) spaces could be obtained in some natural way?

Affirmative answers to these questions motivate the study of cubulated groups—groups which act cocompactly on CAT(0) cube complexes, which are somewhat analogous to simplicial complexes, but with geometry: they are formed by gluing Euclidean cubes along faces by isometries.

Not all CAT(0) groups are cubulated, but constructing an explicit example is a non-trivial exercise (here’s one.)

Cube complexes have hyperplanes, which are (intuition shamelessly stolen from Teddy Einstein) roughly analogous to geodesics; the purely combinatorial structure of the hyperplanes can be used to gain additional insight into the structure of cubulated groups.

Special cube complexes, RAAGs, and Agol’s theorem

The analogy between hyperplanes and geodesics is closer in the case of special cube complexes, which are cube complexes whose hyperplanes do not exhibit any of the four hyperplane pathologies.

Equivalently, though less transparently, special cube complexes are those whose fundamental groups embed in a very special class of groups, the right-angled Artin groups, or RAAGs (these are Artin groups—see below—with the additional stipulation that all relations are commutators.)

In 2012, Ian Agol proved, building on work of Wise and Groves-Manning, that every cubulated hyperbolic 3-manifold group is special, or in other words embeds in a RAAG. In combination with Kahn-Markovic’s resolution of the surface subgroup conjecture and the Haglund-Wise-Sageev construction for cubulating hyperbolic 3-manifold groups, this proved the Virtually Haken Conjecture, and with further work of Agol and Wise involving special cube complex fundamental groups, the Virtually Fibered Conjecture. That proof, in its entirety, involved an extraordinary sweep of ideas, and in part demonstrates the mathematical value of geometric group theory ideas in general and cubulation in particular.

Relative and hierarchical hyperbolicity

There are various other related notions of “broken”, “damaged” or “restricted” negative curvature (none of these being technical terms here) floating around, inspired by various particular examples.

For instance, relative hyperbolicity generalises fundamental group of hyperbolic manifolds with geodesic boundary—these spaces are, roughly speaking, hyperbolic away from certain bad regions—, and hierarchical hyperbolicity is inspired by the refinement of coarse negative curvature seen in the mapping class group following Masur-Minsky theory, where regions of the space may be reasonably modeled by projections to families (“hierarchies”) of Gromov-hyperbolic spaces.


CAT(0) not a quasi-isometry invariant of groups

It is easy enough to produce examples of pairs of spaces which are quasi-isometric, but only one of which is CAT(0): the CAT(k) inequality applies at every scale, but quasi-isometries do not allow any control over what happens at small scales. For instance, we could quasi-isometrically map Euclidean space into a space with regions of (arbitrarily large) positive curvature concentrated inside bounded balls, in which small triangles will fail to satisfy the CAT(0) condition.

It is a little trickier to produce examples of pairs of groups which are quasi-isometric, but only one of which is CAT(0), since now the CAT(0) condition applies not directly to the group, but refers to (the existence of) a CAT(0) space on which the group acts geometrically and cocompactly (and hence a fortiori by semisimple isometries), and we have, at first blush, great freedom to choose what that space might be, depending on what the group is.

Here are two constructions that work, both of which are pointed out in Remark 1 of this paper of Piggott-Ruane-Walsh (although one of them—the second example below—somewhat badly).

Fundamental groups of graph manifolds

One example follows from the work of Kapovich-Leeb on fundamental groups of graph manifolds. Kapovich-Leeb proved (Theorem 1.1) that whenever M is a Haken (i.e. contains an essential properly embedded surface) Riemmanian 3-manifold with \chi(M) = 0, \pi(M) is quasi-isometric to a fundamental group of a compact non-positively curved 3-manifold; groups of the latter description are always CAT(0).

On the other hand, Leeb showed (Example 4.2) that there exist Haken 3-manifolds with \chi(M) = 0—more specifically, closed graph manifolds with no atoroidal components  (i.e. all of whose geometric components are Seifert-fibred) and empty boundary—which do not admit metrics of nonpositive curvature; decomposing the manifold we see that there exist spherical regions in M, and then it follows that the fundamental groups \pi_1(M) of such manifolds M cannot be CAT(0).

Mapping class groups (and unit tangent bundles)

A different example can be found by messing around with mapping class groups; the following is taken from p. 258 of Bridson and Haefliger’s Metric Spaces of Nonpositive Curvature (the top half of the page, not the bottom half.)

Let K be the central extension 1 \to \langle T_c \rangle \to K \to \pi_1(\Sigma_2) \to 1 defined as the kernel of the natural homomorphisms \mathrm{Mod}(\Sigma_2^b) \to \mathrm{Mod}(\Sigma_2^p) \to \mathrm{Mod}(\Sigma_2), where \Sigma_2 is the closed genus-2 surface, \Sigma_2^b denotes the genus-2 surface with a single boundary component c \cong S^1 (and T_c denotes the Dehn twist about c), and \Sigma_2^p denotes the genus-2 surface with a single puncture.

By an unpublished observation of Geoffrey Mess (looking at the geometry of the mapping classes), K is the fundamental group of the unit tangent bundle S\Sigma_2 of \Sigma_2, and hence a cocompact lattice in \widetilde{\mathrm{PSL}_2\mathbb{R}}, but this last does not contain the fundamental group of any closed hyperbolic surface. But, on the other hand, any finite-index subgroup of \pi_1(\Sigma_2) is, by covering space theory, the fundamental group of a closed hyperbolic surface; and now we may conclude that our central extension cannot be split, even after passing to a finite-index subgroup of \pi_1(\Sigma_2).

Now we have the following general result on isometries of CAT(0) spaces: any group of isometries of a CAT(0) space which contains a central A \cong \mathbb{Z}^n subgroup contains a finite-index subgroup which has A as a direct factor.

Hence we conclude that our K cannot be (isomorphic to) a group of semisimple isometries of any CAT(0) space, i.e. K is not a CAT(0) group.

On the other hand, Epstein, Gersten (and Mess? So claims Misha Kapovich) independently proved that \pi_1(S\Sigma_2) is quasi-isometric to \pi_1(\Sigma_2) \times \mathbb{Z}, which is certainly CAT(0), e.g. being the product of two CAT(0) groups. This—that two of the Thurston model geometries, \widetilde{\mathrm{PSL}_2\mathbb{R}} and \mathbb{H}^2 \times \mathbb{R}, are quasi-isometric to each other—is apparently also a [multiply] unpublished observation—trying to track down a reference was annoyingly tricky. Gromov’s Metric Structures for Riemannian and Non-Riemannian Spaces contains an account.

Annotated Definitions

Kähler manifolds

A Kähler metric on a Riemmanian manifold is one which has especially nice (indeed, bemerkenswert) properties, which are usually summarized by saying that it has compatible Riemannian, symplectic, and complex structures. More precisely (and less transparently), a Kähler structure on a Riemannian manifold (M^n, g) is a pair (\Omega, J) where

  • J is a complex structure, i.e. a field of endomorphisms of the tangent bundle TM satisfying J^2 = -\mathrm{id} which is also integrable;
  • \Omega is a symplectic 2-form (i.e. d\Omega = 0);
  • g(X,Y) = g(JX, JY) for all X, Y \in TM (i.e. J makes g a Hermitian metric, or the complex and Riemannian structures play nice with each other), and
  • \Omega(X,Y) = g(JX, Y) (a compatibility condition for the symplectic structure).

For a Kähler metric, the Hermitian metric tensor g is specified by a unique function u, in the sense that g_{\alpha\bar{\beta}} = \frac{\partial^2 u}{\partial z_\alpha \partial \bar{z_\beta}}. This gives rise to simple explicit expression for the Christoffel symbols and the Ricci and curvature tensors, and “a long list of miracles occur then.” (quoted from Moroianu’s notes, which presumably quote from Kähler’s original paper.)

Kähler metrics  may also be characterized, analytically, by the existence of holomorphic normal coordinates around each point.

Some examples of Kähler manifolds (i.e. manifolds which admit Kähler metrics): \mathbb{C}^m with the usual Hermitian metric; any Riemann surface; complex projective space with the Fubini-Study metric; any projective manifold; more examples from algebraic geometry …

Moroianu’s notes contains much more on Kähler manifolds, including elaborations and proofs of many of the assertions above.


Higher Teichmüller Theory

An algebraic viewpoint: character varieties

I have been describing the genus g Teichmüller space Teich(g) as the space of essentially different hyperbolic metrics on the topological surface \Sigma_g of genus g, or the space of conformal structures on \Sigma_g.

It is also possible to give it an altogether more algebraic description, and it is from this viewpoint that the generalisation to “higher” Teichmüller theory is perhaps most easily seen, or at least most superficially obvious.

Recall a point in Teichmüller space is given by a(n equivalence of) pair(s) [(S, h)] where S is a hyperbolic surface of genus g and h: \Sigma_g \to S is an orientation-preserving homeomorphism, which is understood to be an isometry by fiat.

This is equivalent to a choice of isomorphism between the fundamental groups \pi_1(\Sigma_g) \to \pi_1(S). Now \pi_1(S) acts by isometries (deck transformations) on the universal cover of S, which is the hyperbolic plane; thus our choice of isomorphism between fundamental groups gives rise to a representation of the fundamental group \pi_1(\Sigma_g) as isometries of the hyperbolic plane, or, in other words, since the isometry group of the hyperbolic plane is (isomorphic to) \mathrm{PSL}(2,\mathbb{R}), a homomorphism \pi_1(\Sigma_g) \to \mathrm{PSL}(2,\mathbb{R}). These representations are sometimes called holonomy representations.

Thus we can identify Teich(g) with (some subspace of) \mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(2,\mathbb{R})), or rather of a quotient thereof, to take account of the corresponding quotient by homotopy—specifically, a quotient by the conjugation action of \mathrm{PSL}(2,\mathbb{R}).

This quotient \mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(2,\mathbb{R})) / \mathrm{PSL}(2,\mathbb{R}) is often called a representation variety or character variety (although apparently nobody has written down a proof that it is a variety in the algebro-geometric sense; conversely, nobody has a proof that it is not a variety either. It seems like the name came about since \mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(n,\mathbb{C})) / \mathrm{PSL}(n,\mathbb{C}) is [using considerable machinery to handle the quotient] a variety, and also \mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(2,\mathbb{R})) is certainly an algebraic variety by rather more elementary arguments.)

Which part of our character variety is Teichmüller space identified with? We can show that the embedding [(S,h)] \to h_* is open and closed, and hence Teichmüller space is (identified with) a component of the character variety. We also note that holonomy representations are discrete and faithful, and furthermore, using the Margulis lemma, that discreteness and faithfulness are both closed and open conditions, so that the discrete and faithful representations form a connected component of the character variety—this is the component that is identified with Teichmüller space.

We can now ask if anything interesting happens when we replace \mathrm{PSL}_2\mathbb{R} with a different (semisimple) Lie group, possibly of higher rank—this is what the “higher” in “higher Teichmüller theory” most directly refers to. What is the structure of the corresponding character variety? Can we describe any of its connected component in terms of geometric, topological or dynamical properties of interest to us?

Example: Hitchin representations

A Hitchin component is the connected component of \mathrm{Hom}(\pi_1(\Sigma_g), \mathrm{PSL}(n,\mathbb{R})) / \mathrm{PSL}(n,\mathbb{R}) which contains the image of Teichmüller space in \mathrm{Hom}(\pi_(\Sigma_g), \mathrm{PSL}(2,\mathbb{R})) under the irreducible representation \mathrm{SL}(2,\mathbb{R}) \to \mathrm{SL}(n,\mathbb{R}). Elements of the Hitchin component are known as Hitchin representations.

The theory of Hitchin components shares many properties with Teichmüller theory: Hitchin representations are discrete, faithful, and quasi-isometric embeddings; one can prove collar lemmas; there are various coordinate systems on them which generalize coordinates on Teichmüller space.

[[ future addition: Higgs bundles and “deep connections” with algebraic geometry, which I do not understand but would like to [at least a little]. ]]

Why surface (or 3-manifold) group representations into semisimple Lie groups?

From MathOverflow: “… the universal cover together with the deck group action contain a lot of information about the manifold, and the representations of the group provide one way to extract it … The space of representations into $latex \mathrm{SL}(n,\mathbb{C})$ is naturally an algebraic variety equipped with an additional rich structure which can conceivably be used to produce invariants of the original manifold.”

A geometric viewpoint: (G,X)-structures

We could also ask if (components in) our new character varieties \mathrm{Hom}(\pi_1(\Sigma), G) / G parametrize geometric structures of any sort. Here we are implicitly (or perhaps to some extent explicitly) using a point of view first expressed in Klein’s Erlangen program and nowadays formulated using the notion of (G, X)-structures, in which a “geometry” is characterized primarily by its symmetries—or, more precisely, described by a connected, simply-connected manifold X together with a Lie group G of diffeomorphisms acting transitively on X with compact point stabilizers.

Thus for instance Euclidean geometry is described by (\mathrm{SL}_n \mathbb{R} \ltimes \mathbb{R}^n, \mathbb{R}^n), or hyperbolic geometry by (\mathrm{SO}(n,1), \mathbb{H}^n)).

Thus if we can identify our Lie group G in the character variety \mathrm{Hom}(\pi_1(\Sigma), G) / G as acting transitively with compact point stabilizers on some connected, simply-connected manifold X, we will have a description of (at least certain components of) the character variety as parametrizing (certain) (G,X)-structures on the surface (or 3-manifold, or n-manifold for higher n, though what is known as n increases diminishes very rapidly.)

For instance: when G was \mathrm{PSL}_2\mathbb{R} \cong \mathrm{Isom}^+(\mathbb{H}^2), we could take X = \mathbb{H}^2, and the corresponding character variety—or rather the component thereof which consisted of discrete, faithful representations—, which, recall, is exactly Teichmüller space, then parametrizes hyperbolic structures on the surface \Sigma. Aha.

Example: the third Hitchin component and convex real projective structures

\mathrm{SL}_3 \mathbb{R} (or rather the central quotient \mathrm{PSL}_3 \mathbb{R}) is the automorphism (isometry) group of real projective space \mathbb{RP}^2, and indeed Choi and Goldman proved that the n = 3 Hitchin component parametrizes convex real projective structures on a surface.

Danny Calegari exposits at more length on this moduli space on his blog.

In general, though, it is not so easy to obtain descriptions of higher representation varieties—even of the Hitchin components with n \geq 4—in terms of intuitively-comprehensible (G,X)-structures; in that respect it is an open question to obtain descriptions of such a geometric flavour.

(G,X)-structures and flat bundles

A (G,X)-bundle on a manifold M is a space E together with a (projection) map E \to M whose fibers are homeomorphic to X, and which admits local trivialisations with transition maps in G. For example: (G,X)-bundles where X is a vector space k^n and G = mathrm{GL}(n,k) are vector bundles.

(G,X)-bundles are, from one perspective, yet another way of globally encoding a collection of locally X-like structures patched up by bits of G, and indeed we can systematically go between them and (G,X)-structures by taking into account two additional pieces of information:

  1. flat connection, which may be visualized as a “horizontal” foliation transverse to the fibers, which are preserved by the transition / gluing maps on M, and
  2. a section, i.e. a left inverse to the projection map, transverse to the fibers.

Given a (G,X)-structure on M, we have a flat (G,X)-bundle on M with fibers isomorphic to X and local trivialisations described by the (G,X)-structure charts X \times U_i, with flat connection described by the horizontal foliation on X \times X, together with a section—the diagonal section of X \times X—transverse to the foliation.

Conversely, given a flat (G,X)-bundle on M (a bundle equipped with such a flat connection is known as a flat bundle) together with a section of the bundle transverse to the fibers, we can effectively reverse the above process to obtain a (G,X)-structure on M: intuitively speaking, the flat connection helps us determine where M is inside the total space of the bundle, and the section specifies which bit of X locally models each region of M.

Example: maximal representations

A rather different way of picking out a component of interest starts with Milnor’s observation, subsequently extended by Wood to the Milnor-Wood inequality, that the Euler number of any flat plane bundle over a hyperbolic surface \Sigma is at most -\chi(\Sigma) in absolute value. Goldman, in his doctoral thesis, proved that the representations \pi_1(\Sigma) \to \mathrm{PSL}_2\mathbb{R} whose associated flat (\mathrm{PSL}_2\mathbb{R}, \mathbb{H}^2)-bundle has maximal Euler number -\chi(\Sigma) are precisely those which are holonomy representations of hyperbolic structures.

In other words, the maximal level set of the Euler number invariant in this case is a component in the character variety (Teichmüller space) of geometric interest.

Motivated by this, we may consider other representation invariants, often similarly constructed using cohomology, and define maximal representations as representations in the maximal level set of these invariants, where bounds for these invariants analogous to the Milnor-Wood inequality exist and where the level sets are well-behaved.

[[ future addition: some actual examples ]]

These maximal representations of surface groups have been shown to have some good geometric and dynamical properties: for instance, they are discrete and faithful; they are quasi-isometric embeddings when they are representations of closed surface groups; they are Anosov (see below.)

 Another algebraic viewpoint: lattices

Fundamental groups of closed or indeed finitely-punctured surfaces (with genus at least 2) are lattices in \mathrm{Isom}(\mathbb{H}^2) \cong \mathrm{PSL}_2 \mathbb{R}; similarly fundamental groups of closed, or more generally finite-volume hyperbolic 3-manifolds are lattices in \mathrm{Isom}(\mathbb{H}^3) \cong \mathrm{PSL}_2 \mathbb{C}.

We can thus view our character varieties as spaces of representations of lattices into semisimple Lie groups, and ask what happens, in slightly greater generality, as we vary the Lie group/s from which we take our lattices and into which we our representations take them. Here there is a striking contrast between what happens in low rank / dimension, and what happens in higher dimension / rank.

Hyperbolic surfaces carry a great multiplicity of possible hyperbolic structures and deformations: a whole Teichmüller space’s worth of them. On the other hand, finite-volume hyperbolic 3-manifolds are extremely rigid: Mostow-Prasad rigidity states that any homotopy equivalence between finite-volume hyperbolic 3-manifolds is induced by an isometry. Even stronger rigidity results hold for higher-rank Lie groups: Margulis superrigidity states that, loosely speaking, any linear representation of an irreducible lattice in a higher-rank semisimple Lie group is induced by a representation of the ambient Lie group. In other words, the deformation spaces of such lattices are trivial.

The main moral of the story here seems to be that—to speak in imprecise terms for a moment—, should there still be any geometric structures that we can associate to the points in our character varieties of 3-manifold or higher representations, we should not expect them to be closed, finite-volume, or similarly tame.

Dynamical developments: Anosov representations

The examples of higher Teichmüller spaces above, somewhat disparate though they may be, share certain common structures, first explicitly described by Labourie for Hitchin representations, and subsequently systematically developed for more general representations \Gamma \to G of word-hyperbolic groups \Gamma into semisimple Lie groups G in Guichard-Wienhard.

Roughly speaking, these structures may be described as pairs of transverse limit maps which pick out attracting and repelling spaces at each point on \partial_\infty \Gamma, the Gromov boundary of our word-hyperbolic group, in a continuous way. This in turn gives rise to a coarsely Anosov structure (not a technical term—the right technical term here being either “metric Anosov flow” or “Smale flow”, depending on whom you ask) on the Gromov geodesic flow, which provides the setting for dynamical arguments to further the geometry of our representation varieties.

Cartan projections, dominated splittings, and domains of discontinuity

[[ future addition: subsequent work of G(GK)W and KLP / see also Bochi-Potrie-Sambarino. Include Wienhard’s description of positive, maximal, and “mixed” representations? ]]

This gives rise to an identification of representations in (the specified components of) our character varieties as holonomies of certain geometric structures. There appear to be considerable difficulties involved, however, in attempting to make the converse of such an identification effective, i.e. in determining whether a given representation is a holonomy of a geometric structure of the type in question.

Pressure metrics

Given a space \mathcal{M} of Anosov representations, we can associate to each representation \rho \in \mathcal{M} a Hölder function f_\rho given by a natural reparametrisation function for the Gromov geodesic flow associated to the representation.

There is now a natural dynamical invariant, the pressure, on the space of Hölder functions, which by some fairly heavy machinery from dynamics (the thermodynamic formalism) varies analytically; moreover, to each Hölder function f_\rho above there is naturally associated a pressure-zero function h(f_\rho) f_\rho, where h(f_\rho) is the topological entropy of the flow associated to f_\rho.

Now, again by the thermodynamic formalism, the Hessian of the map \rho \mapsto h(f_\rho) f_\rho is a well-defined positive-semidefinite quadratic form on the representation variety \mathcal{M}. With considerably more work, Bridgeman-Canary-Labourie-Sambarino showed that it is in fact positive-definite, and hence defines a Riemannian metric on the representation variety in question.

When \mathcal{M} is Teichmüller space, the pressure metric is equivalent to the Weil-Petersson metric, about which some things are known (although many are not.) In more general cases, the geometry of the pressure metric is a wide-open question and an area of active research. How can we describe geodesics in this metric? Is it complete, and if not what is its completion? Is the metric, in general, negatively-curved?


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.


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.


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.


Deformation spaces and Teichmüller theory

So we have this thingummy, and we’re trying to find out more about it. What would you do? Shake it a little and see what happens, maybe? That’s one way you might describe the motivation for (or, at least, one motivation for) deformation theory. By figuring out how a geometric structure—a triangulation, say, or a foliation, or a smooth atlas, or a hyperbolic structure—can be perturbed, we learn more about the nature of the structure at hand.

Deformation spaces are not a precisely-defined concept; moduli spaces are slightly more precisely-defined, though not completely so, and in some sense a similar (though perhaps somewhat broader) idea.

One key example of a moduli space is the genus g Teichmüller space Teich(g): this is the space of all marked hyperbolic metrics on a genus g surface \Sigma_g (the genus g surface, when we regard it as the unique topological object in its homeomorphism class), modulo homotopy.

Here “marked” indicates that points in Teich(g) correspond not just to a choice of hyperbolic metric on \Sigma_g (described mathematically by e.g. an orientation-preserving homeomorphism from a “standard” / fixed surface with a metric which is declared to be an isometry), but also a choice of isomorphism to the fundamental group \pi_1(\Sigma_g)—not just hyperbolic clothing, but also instructions on how to wear it. The specification of a marking “rigidifies” and helps clarify the effect of automorphisms (i.e. non-trivial mapping classes) on our hyperbolic surface—something which is concern e.g. when we try to build a fine moduli space.

It is one of the great triumphs of 19th century mathematics that Teich(g) is also the space of all marked conformal structures on the genus g surface, modulo biholomorphisms homotopic to the identity. Any conformal structure yields a hyperbolic metric via uniformization; conversely, given a hyperbolic metric, we may produce a conformal structure using isothermal coordinates.

This dual identity allows Teichmüller theory—the theory which describes Teichmüller space and what it can do—to draw on tools from both hyperbolic geometry and complex analysis, and gives the theory much of its richness.

Setting out from the viewpoint of hyperbolic geometry, we may obtain, for instance, the Fenchel-Nielsen coordinates—which start by taking all-right hyperbolic hexagons, gluing these to get pairs of pants with specified cuff lengths, and then gluing pairs of pants with specified twists to get hyperbolic surfaces—and a proof of the Nielsen-Thurston classification, parts of which look at how lengths of curves on the hyperbolic surface change under the action of (various types of) mapping classes.

Setting out from the viewpoint of complex analysis, we obtain a systematic description of the tangent and cotangent spaces to Teichmüller space as spaces of various sorts of differential forms—Beltrami differentials and holomorphic quadratic differentials, respectively—, which in some sense encode representations of infinitesimal deformations of the metric structure—quasiconformal maps and transverse measured foliations, resp.

From there we may obtain results on optimal deformations between the various hyperbolic / conformal structures—this is the content of Teichmüller’s existence and uniqueness theorems, and the basis of Teichmüller geometry, i.e. the study of the geometry of Teich(g) as not just a collection of spaces, but a space in its own right.

I’ve written elsewhere, at some [mild] length, about moduli spaces in general and Teichmüller space in particular, including the motivation for these, so I should not continue on here. I did want, nevertheless, to emphasise the dual nature of the space and the tools involved—something I’ve come to appreciate more recently—, and to highlight the theory’s role as a base case and starting point for some of the subsequent areas I explore next: higher Teichmüller theory, and Weil-Petersson geometry.

Overview / Outlines

This Corner of the Field

Over the next four months or so*, I will be writing a series of blogposts which outline in broad, descriptive terms the area/s of mathematics I am most interested in—a large swathe of low-dimensional topology and its environs, for the most part—, some sets of tools which are useful to these areas, and how they relate to one another and to the rest of mathematics and to other bits of human inquiry.

The purpose of (writing) these posts is mostly to seek an at least mildly coherent answer to the question, “so what is it you do?” and the often-unasked but—at least to me—entirely natural follow-up, “but why?”

In order to do this I will attempt to describe just what each of these bits of mathematics is about, how it came about, why its originators found it interesting, and why I find it interesting. Along the way, as our dear doctoral chair would advise, there should be plenty of concrete illustrative examples. In some cases, I may also describe current directions of exploration and active research.

*A rough deadline … or more of a broad scheduling guideline. To some extent, I’m inclined to treat [large parts of] this as an ongoing project, with posts—particularly ones covering areas I am particularly focused on / interested in—subject to active and ongoing revision.

An outline of the ground to be covered, to be updated with links and edits as I go along: