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.