Simplicial complexes defined on groups

TL;DR

This paper explores extending graph-based group structures to simplicial complexes, defining various complexes preserved by group automorphisms and examining their relation to known graphs and independence complexes.

Contribution

It introduces new simplicial complexes on groups, analyzes their properties, and investigates conditions under which certain complexes coincide, expanding the understanding of group-related topological structures.

Findings

Two forms of independence complexes are deeply connected to graph structures.

Conditions identified for when these complexes coincide in certain groups.

Several new simplicial complexes on groups are characterized.

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.