Simple Expansion Sets and Non-Positive Curvature

TL;DR

This paper introduces simple expansion sets that lead to the construction of non-positively curved cubical complexes, providing new insights into the actions of various Thompson and Houghton groups on CAT(0) spaces.

Contribution

It establishes a systematic method to construct CAT(0) cubical complexes from simple expansion sets, unifying proofs for group actions on such spaces.

Findings

Simple expansion sets determine CAT(0) cubical complexes.

Thompson groups and Houghton's groups act on CAT(0) complexes.

Conditions for local finiteness of the complexes are identified.

Abstract

An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups. We say that $\mathcal{B}$ is simple if proper expansions are unique when they exist. We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompsons groups $F$, $T$, and $V$, Houghton's groups $H_{n}$, and groups defined by finite similarity structures all act on CAT(0) cubical complexes. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of $F$, $T$, $V$, and $H_{n}$.