Finiteness Properties of Locally Defined Groups

TL;DR

This paper introduces a unified framework for constructing geometric models and studying the finiteness properties of a broad class of groups generated by locally defined partial bijections, including many well-known groups.

Contribution

It provides a general construction method for geometric models of these groups and a unified approach to analyze their finiteness properties.

Findings

Unified geometric models for a wide class of groups.

Framework applicable to groups like Thompson's V, Nekrashevych-Röver, Houghton's, and Brin-Thompson groups.

Enhanced understanding of finiteness properties across these groups.

Abstract

Let $X$ be a set and let $S$ be an inverse semigroup of partial bijections of $X$. Thus, an element of $S$ is a bijection between two subsets of $X$, and the set $S$ is required to be closed under the operations of taking inverses and compositions of functions. We define $\Gamma_{S}$ to be the set of self-bijections of $X$ in which each $\gamma \in \Gamma_{S}$ is expressible as a union of finitely many members of $S$. This set is a group with respect to composition. The groups $\Gamma_{S}$ form a class containing numerous widely studied groups, such as Thompson's group $V$, the Nekrashevych-R\"{o}ver groups, Houghton's groups, and the Brin-Thompson groups $nV$, among many others. We offer a unified construction of geometric models for $\Gamma_{S}$ and a general framework for studying the finiteness properties of these groups.