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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.

Quasi-isometry

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.

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