Finiteness properties of locally defined 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–Röver 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.