Finiteness properties of locally defined groups

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 ₒ to be the set of self-bijections of X in which each ₒ is expressible as a union of finitely many members of S. This set is a group with respect to composition. The groups ₒ 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 ₒ and a general framework for studying the finiteness properties of these groups.