Groups and graphs (and computation)

Graphs in group theory

Given a group G, or more specifically a finite set S of generators for G, one can form the Cayley graph \mathrm{Cay}(G,S) by taking the vertex set to be the set of group elements, and adding an edge vw iff v = ws for some s \in S.

One can then attempt to study algebraic properties of the group by looking at properties of (one of) its Cayley graph(s). In the case of an infinite group G, one can show that any two Cayley graphs, i.e. the Cayley graphs associated to any two sets of generators, have the same coarse geometry (in the sense of quasi-isometry—see the post on geometric group theory), so that the coarse geometric properties of any Cayley graph can be considered to be intrinsic geometric properties of the group, rather than properties of any particular presentation thereof. Indeed, this is the key insight that drives geometric group theory.

One particularly striking example where such geometric methods seem to yield great insight with much less pain than directly algebraic ones is in the case of free groups and their automorphism groups, which may be thought of geometrically as fundamental groups of bouquets of circles. This is equivalent to the Cayley graph viewpoint: the Cayley graph is the universal cover of the bouquet of circles. Automorphisms of the free group then correspond to homotopy equivalences of the bouquet of circles, and arguments about the automorphism group can be conducted in terms of these homotopy equivalences, leading to such results as classifications and characterizations of these automorphisms analogous to the classification of mapping classes of surfaces.

The same arguments apply to finite groups, although in this case the graphs are finite, and so quasi-equivalent to trivial graphs (i.e. the graph with one vertex and no edges), and geometric group theory has nothing more to say. From this point of view, finite groups are, in a very precise sense, virtually trivial. I do wonder though: can the theory of finite graphs, itself a rather rich area of combinatorics, be fruitfully used to say / argue interesting things about finite groups?

Graphs also appear in group theory in the notion of graphs of groups which is fundamental to the Bass-Serre theory of groups acting on simplicial trees. Here the graph acts essentially as a book-keeping mechanism to record the structure of the group as built up from vertex and edge stabilisers; these graphs have also been described as “one-dimensional analogues of orbifolds”. Group actions on trees appear in a variety of settings, for fundamental reasons I do not yet fully understand, and so this notion takes on some importance.

Groups in graph theory

Conversely, group theory can play a role in elucidating the structure theory of graphs. For instance, group theory played a significant role in Laci Babai’s recent breakthrough on graph isomorphism. Very roughly speaking, Babai’s algorithm relies on the heuristic that any isomorphism between graphs has to preserve degree. Thus, given a highly irregular graph, a brute-force search could actually be (sufficiently) efficient, since the number of possible isomorphisms is small relative to the number of vertices.

The problem, then, is if we have highly regular graphs. One way around this is to consider an equitable partition; but one may not always have one of these either. However, Babai proved that the only obstacle to the existence of such a partition is the existence of a large Johnson graph, and then used subtle structural results, of an essentially algebraic nature, on Johnson groups and how they sit inside larger (finite) permutation groups to obtain, constructively, either an isomorphism, or a certificate of non-isomorphism (a provably empty automorphism group.)

In the language of Babai’s preprint: “We shall certify both the presence and the absence of local symmetry. … If we don’t find local symmetry, we infer global irregularity; if we do find local symmetry, we either build it up to global symmetry (find automorphisms) or use any obstacles to this attempt to infer local irregularity which we then synthesize to global irregularity. Building up symmetry means approximating the automorphism group from below; when we discover irregularity,we turn that into an effective upper bound on the automorphism group . When the lower and the upper bounds on the automorphism group meet, we have found the automorphism group and solved the isomorphism problem.”

Here the group theory plays a key role in introducing more rigidity / structure to an otherwise fairly flexible and fluid structure. The structural theory of finite groups is used in a fairly strong way: indeed, the proof relies on the classification of finite simple groups, although the strict logical dependence is only on one of its corollaries (Schreier’s statement that the outer automorphism group of a finite simple group is solvable.)

A concrete application to coding theory

Group theory and graph theory both come into play in e.g. the construction and study of codes from Cayley graphs and Schreier graphs: the algebraic structure introduced by the group-theoretic construction of these graphs is very helpful in proving key desired properties of these codes.