  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.
