Simplicial complexes defined on groups

1 Introduction

The last couple of decades have seen a big upsurge in activity about graphs defined on groups. In my opinion, the most important aspects of this study are as follows:

• defining classes of groups by properties of graphs defined on them;

• finding new results about groups using graphs;

• constructing beautiful graphs from groups.

When I lecture about this topic, it often happens (most recently at the joint American Mathematical Society/Unione Matematica Italiana meeting in Palermo in July 2024) that someone asks, “What about simplicial complexes?” This is a natural question, since there are properties which are not determined by pairwise relations on the group, but needs to be handled with some care since, for example, generating sets for a group do not form a simplicial complex.

A simplicial complex Δ is a downward-closed collection of finite subsets (called simplices or simplexes) of a set X . We assume that every singleton of X belongs to Δ . For geometric reasons, a simplex of cardinality k has dimension k − 1 . Thus a point or vertex of X has dimension 0, while an edge { x , y } has dimension 1, and a triangle { x , y , z } has dimension 2.

The k-skeleton of Δ consists of all the subsets of dimension at most k (thus, cardinality at most k + 1 ). Thus, the 1-skeleton, is a graph.

Graph theory is a subject with centuries of history, and many parameters and properties of graphs have been studied, including cliques and cocliques, colourings, matchings, cycles, and spectral properties. A glance at a survey of just one class of graphs defined on groups, the power graph [1], shows this very clearly.

On the other hand, simplicial complexes are primarily of interest to topologists: triangulating a topological space gives a simplicial complex, whose properties converge to those of the space as the triangulation becomes finer (for nice spaces). There has been relatively little work on simplicial complexes in their own right, or combinatorial studies of them. Two of the rare exceptions are Euler complexes [2], which arise in game theory, and Ramanujan complexes [3], a higher dimensional version of Ramanujan graphs, which have strong expansion properties.

I conclude this introduction with a brief description of some of the graphs on a group G , which play a part in this story [4]. Many of these can be defined in terms of a subgroup-closed class C of groups; usually we assume that C contains all cyclic groups. In the corresponding graph, x and y are joined if and only if ⟨ x , y ⟩ ∈ C . Examples for C , and the corresponding graphs, include

• cyclic groups (the enhanced power graph);

• abelian groups (the commuting graph);

• nilpotent groups (the nilpotency graph);

• soluble groups (the solubility graph).

• the power graph: x and y are joined if one is a power of the other;

• the generating graph: x and y are joined if ⟨ x , y ⟩ = G .

• the independence graph: x and y are joined if they are contained in a minimal (under inclusion) generating set;

• the rank graph: x and y are joined if they are contained in a generating set of minimum cardinality.

To conclude the introduction, here are two general questions which can be asked of a simplicial complex on a group:

2 Complexes defined by independence

I begin with two complexes on a group defined in terms of independence. A subset A of a group G is independent if none of its elements can be expressed as a word in the other elements and their inverses; or equivalently, if a ∉ ⟨ A \ { a } ⟩ for all a ∈ A .

3 Further examples

Let C be a subgroup-closed class of groups. Then, the collection of subsets S of a group G for which ⟨ S ⟩ ∈ C is a simplicial complex, for which every element of G is a point if C contains all cyclic groups. The 1-skeleton of this complex is the graph defined in terms of C in the introduction.

The maximal simplexes are precisely the maximal C -subgroups of G . So no maximal simplex generates G unless G ∈ C , in which case G is a simplex.

Furthermore, if the class C has the property that a group belongs to C if and only if its 2-generator subgroups belong to C , then a subset, which induces a complete subgraph of the 1-skeleton, is a simplex. This holds, for example, if C is the class of cyclic groups, or the class of abelian groups.

The generating sets do not form a simplicial complex, but their complements do: this is the non-generating complex.

Further simplicial complexes can be formed by imposing two different conditions on the simplices. Examples include

• the commuting independence complex, where a simplex is an independent set whose elements commute pairwise;

• the non-generating independence complex, where a simplex is an independent set, which does not generate the group.

These somewhat resemble various difference graphs, which have been studied on groups, such as the difference between the enhanced power graph and the power graph [11]. I mention here also the work of Saul Freedman on the non-commuting, non-generating graph [12], and the work of Freedman et al. on groups for which the power graph is equal to the complement of the independence graph, or the enhanced power graph is the complement of the rank graph [5].

Various suggestions for simplicial complexes have, roughly speaking, the form: { x 1 , … , x k } is a simplex if and only if w ( x 1 , … , x k ) = 1 , for some fixed word w . There are a couple of problems with this definition as stated. First, it depends on the order of x 1 , … , x k ; so, instead, we have to say, { x 1 , … , x k } is a simplex if and only if there is a permutation π ∈ S k such that w ( x 1 π , … , x k π ) = 1 . Also, we have to insist that x 1 , … , x k are all distinct.

Then, the problem arises of extending the definition to other dimensions. Going up is easy: { x 1 , … , x m } is a simplex (for m ≥ k ) if and only if all of its k -subsets are simplices. For going down, we have to say, { x 1 , … , x l } is a simplex, for l ≤ k , if and only if there exist x l + 1 , … , x k such that { x 1 , … , x l , x l + 1 , … , x k } is a simplex.

A simple example with k = 2 is the commuting complex described earlier, with w = [ x , y ] = x − 1 y − 1 x y . Examples with k = 3 include w ( x , y , z ) = x y z , or z ( x , y , z ) = [ [ x , y ] , z ] . In these cases, some partial symmetries already hold, so we do not need to apply the condition for all π ∈ S k . We have

and

I will not discuss these examples further.

4 Gruenberg-Kegel complex

The Gruenberg-Kegel graph of a finite group G is the graph whose vertex set is the set Π of prime divisors of ∣ G ∣ , with p and q joined if G contains an element of order p q . It was introduced by Gruenberg and Kegel in connection with the integral group ring of G . They determined the groups for which the Gruenberg-Kegel graph is disconnected, but did not publish their result. It was published by Gruenberg’s student Williams [13]. These graphs have been the subject of a lot of further research.

Define the Gruenberg-Kegel complex GKC ( G ) of G by the rule that the vertex set is Π (as above), and a subset Γ is a simplex if and only if G contains an element whose order is the product of the primes in Γ .

For a recent survey on the Gruenberg-Kegel graph [9].

5 Homology and representations

A simplicial complex possesses homology (and cohomology) groups. For complexes defined on G and invariant under Aut ( G ) , these will be Aut ( G ) -modules.

Homological algebra has been around for a while, so perhaps little new can be found about homology groups. On the other hand, persistent homology is an important new tool in data analysis, so maybe homological tools have a role to play here.