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Simplicial complexes defined on groups

Peter J. Cameron

Abstract

This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the group, and in many cases have a relation to familiar graphs on the group. The ones which seem to reach deepest into the graph structure are two forms of independence complex, and some results on the class of groups for which these two complexes coincide are given. Other examples are treated more briefly.

Simplicial complexes defined on groups

Abstract

This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the group, and in many cases have a relation to familiar graphs on the group. The ones which seem to reach deepest into the graph structure are two forms of independence complex, and some results on the class of groups for which these two complexes coincide are given. Other examples are treated more briefly.

Paper Structure

This paper contains 7 sections, 3 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Complexes defined by independence
  3. The independence complex
  4. The strong independence complex
  5. Further examples
  6. The Gruenberg--Kegel complex
  7. Homology and representations

Key Result

Theorem 2.2

An independent set in the symmetric group $S_n$ has size at most $n-1$, with equality if and only if it is also a generating set.

Theorems & Definitions (8)

  • Proof 1
  • Theorem 2.2
  • Proof 2
  • Theorem 2.4
  • Proof 3
  • Theorem 2.5
  • Proof 4
  • Proof 5