Groups from geometry
Besides groups which have geometry, there are also groups with various geometric origins, and these make up another broad vein of the federated entity that is geometric group theory. Often these geometric origins allow us to give the groups themselves a certain geometry; but they also give us directly geometric tools for studying these groups.
The first examples we might think of include surface groups and hyperbolic 3-manifold groups, and more generally fundamental groups of geometric objects. And then we start thinking more …
Mapping class groups
The mapping class group Mod(S) of a surface S of finite type is the group of all homeomorphisms from the surface to itself, mod homotopy. These homeomorphisms are required to fix the boundary pointwise, and preserve the set of punctures setwise (i.e. mapping classes may permute punctures.)
The notation originates with Fricke, who called the “automorphic modular group”, by analogy with the classical modular group which indeed is the mapping class group of the torus.
Mapping classes represent essential symmetries of the topological surface. Mapping class groups can be computed, in general, using the Alexander method—see e.g. Section 2.3 of Farb and Margalit’s Primer.
A large class of easily visualizable mapping classes is given by Dehn twists. There are many mapping classes which are not Dehn twists, but it turns out that mapping class groups are generated by a finite number of (judiciously-chosen) Dehn twists.
One proof of this proceeds by looking at the action of the mapping class group on a (slightly-modified) curve complex. The mapping class group acts on Teichmüller space by changing the marking—the mapping class sends the point to —and also on the curve complex and its relatives, by extending the obvious action on the 1-skeleton (curves go to curves, and we can check that adjacency relations in the curve complex are preserved.)
Looking at these two group actions also gives us proofs that mapping class groups are finitely-presented, residually finite, satisfy a subgroup alternative, and so on.
Note that these two actions complement each other, in the sense that Teichmüller space is compact(ifiable) but not (coarsely) hyperbolic, whereas the curve complex is not locally-compact but is coarsely hyperbolic. Playing these off against each other is key in one proof of the subgroup alternative, for instance.
Outer automorphism groups of free groups
Given any group in general, the outer automorphism group consists of all the automorphisms of , modulo inner automorphisms.
Outer automorphism groups of (finitely-generated, non-abelian) free groups have a particularly geometric flavour, since they may be thought of in terms of their action on a graph with fundamental group , i.e. (or e.g.) a bouquet of n circles.
This leads us to the construction of Culler-Vogtmann outer space , a space of marked metric graphs, whose points are specified by pairs where X is a metric graph with fundamental group (isomorphic to) and is an isomorphism of fundamental groups, and the square braces indicate that we are quotienting out by homotopy equivalences.
This appears very analogous to the definition of Teichmüller space, and indeed there is an analogy between Teichmüller theory and the theory of which, although somewhat informal and imperfect, has driven and illuminated much of the development of the latter.
Similar perspectives may be brought to bear on the study of wherever is naturally the fundamental group of some nice class of geometric or topological objects, e.g. if is a RAAG (which is the fundamental group of its associated Salvetti complexes.)
Coxeter groups and Artin groups
Coxeter groups are groups which generalize discrete reflection groups. The finite Coxeter groups are precisely the finite Euclidean reflection groups, and indeed any Coxeter group may be realized as a discrete reflection group
Coxeter groups have presentations of the form , where relations of the second sort equal numbers of generators (alternating between and ) on both sides. The information needed to write this representation (or a representation for the corresponding Artin group, see below) may be encoded in the form of a finite graph, known as Dynkin diagrams.
Artin groups are also known as generalized braid groups, a braid group being (one way to define it being) the mapping class group of a disk with finitely many punctures. Braid groups have presentations of the form . More generalized Artin groups are given by presentations of the form , where the relations are of the same form as those which appear above in presentations for Coxeter groups.
We note that these presentations are very similar—the only difference being the involutive relations for the generators in a Coxeter group presentation—, and indeed to every Coxeter group there is an associated Artin group (and vice versa) obtained by removing (or adding, resp.) those involutive relations.
Proceeding along this line of thought leads to a geometric interpretation of Artin groups: whereas a Coxeter group W is a group of reflections in some finite hyperplane arrangement, the corresponding Artin group A is the fundamental group of the corresponding complexified hyperplane arrangement modulo the action of the Coxeter group. Moreover, at least for W finite, is aspherical, so it is in fact a ; the same is conjectured to be true for more general W,
Artin groups associated to finite Coxeter groups (called spherical Artin groups) sharmany properties of braid groups. Non-spherical Artin groups (i.e. those whose associated Coxeter groups are infinite) are much less well-understood, but are conjectured to share many of the same good algorithmic / geometric properties of their spherical cousins.
Lie groups and lattices
A different class of examples comes from remarking that fundamental groups of closed manifolds are lattices in Lie groups, and asking about more general lattices in Lie groups.
This allows us to leverage (algebraic) tools from Lie theory to use alongside our geometric tools to study our examples.
[[ future addition: examples in lower rank / rigidity and superrigidity results in higher rank ]]
Dynamical perspectives: Property (T) and amenability
A third major vein of the not-quite-a-single-subject involves using dynamics to study group actions and what they say about groups. Here (at least) two things stand out.
One is Kazhdan’s property (T): a group has (T) if any unitary representation of a group which has an almost-invariant vector has an invariant vector. Equivalently, [[long list here]]
[[ future addition: equivalent characterizations of (T) and properties of (T) groups ]]
Kazhdan proved that lattices in Lie groups have (T) iff the ambient Lie groups have (T). Simple Lie groups of higher rank have (T), and thus for , has (T). His motivation for all of this was apparently to demonstrate the finite generation of these lattices (it can be shown that any discrete countable group with (T) is finitely-generated.)
It seems a bit of a crazy detour, but perhaps entirely reasonable if one is well-versed and comfortable in the language of unitary representations.
On the other hand, there is amenability: [[ future addition: (at least part of) long list of equivalent definitions ]].
Amenable groups are, from a dynamical perspective at least, similar to abelian groups—many dynamical arguments that work for abelian groups can be made to work, in some form, for amenable groups.
[[ future addition: some properties of amenable groups ]]
Amenable groups with property (T) are compact, and in some sense amenability and (T) are opposite: amenability makes almost-invariant vectors easy to find, whereas (T) makes them hard to find.