On groups and simplicial complexes

TL;DR

This paper develops a framework linking $k$-regular simplicial complexes to group theory, generalizing concepts from regular graphs, and introduces a universal arboreal complex as a foundational structure.

Contribution

It introduces a new approach to study $k$-regular simplicial complexes via a universal arboreal complex, extending group-theoretic methods to higher dimensions.

Findings

Answered a question on spectral gaps of higher dimensional Laplacians.

Proved a high dimensional analogue of Leighton's graph covering theorem.

Suggested a random model for $k$-regular $d$-dimensional multicomplexes.

Abstract

The theory of $k$-regular graphs is closely related to group theory. Every $k$-regular, bipartite graph is a Schreier graph with respect to some group $G$, a set of generators $S$ (depending only on $k$) and a subgroup $H$. The goal of this paper is to begin to develop such a framework for $k$-regular simplicial complexes of general dimension $d$. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of $k$-regular simplicial complexes as quotients of one universal object: the $k$-regular $d$-dimensional arboreal complex, which is itself a simplicial complex originating in one specific group depending only on $d$ and $k$. Along the way we answer a question from [PR12] on the spectral gap of higher dimensional Laplacians and prove a high dimensional analogue of Leighton's graph covering theorem. This approach also suggests a random model for $k$-regular $d$-dimensional multicomplexes.