2020/10/15 by Daniel Farley, Farley, Daniel S., Bruce Hughes +1
Mathematics · #20F65 #20J05 #20M18 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2010.08035
openalex publication_date 2020/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ΓS to be the set of self-bijections of X in which each γ∈ ΓS is expressible as a union of finitely many members of S. This set is a group with respect to composition. The groups ΓS 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 ΓS and a general framework for studying the finiteness properties of these groups.