Geometry of Houghton's Groups

Abstract

Ken Brown showed finiteness properties of Houghton's groups by studying the action of those groups on infinite dimensional cell complex. We modify his proof by construction finite dimensional CAT(0) cubical complexes on which Houghton's groups act. We extend D. L. Johnson's result about finite presentation for basic case of Houghton's group to get finite presentations for all Houghton's groups beyond the base case. We also provide exponential isoperimetric inequalities for Houghton's groups.