Simple expansion sets and non-positive curvature - Farley, Daniel (Topology Proceedings, 2025)
An expansion set is a set B such that each b B is equipped with a set of expansions E(b). The theory of expansion sets offers a systematic approach to the construction of classifying spaces for generalized Thompson groups, as described in [6]. We say that B is simple if proper expansions are unique when they exist, or, equivalently, if |E(b)| 2 for all b B. We will prove that any given simple expansion set determines a cubical complex with a metric of non-positive curvature. In many cases, the cubical complex will be CAT(0). We are thus able to recover proofs that Thompson’s groups F, T, and V[7,8], Houghton’s groups H_n, and groups defined by finite similarity structures [9,11,14] all act on CAT(0) cubical complexes with finite stabilizers. We further state a sufficient condition for the cubical complex to be locally finite, and show that the latter condition is satisfied in the cases of F, T, V, and H_n.