Geometry of Houghton’s Groups

Abstract. Hougthon’s groups {Hn} is a family of groups where each Hn consists of ‘trans-lations at infinity ’ on n rays of discrete points emanating from the origin on the plane. Brown shows Hn has type FPn−1 but not FPn by constructing infinite dimensional cell complex on which Hn acts with certain conditions. We modify his idea to construct n-dimensional CAT(0) cubical complex Xn on which Hn acts with the same conditions as before. Brown also shows Hn is finitely presented provided n ≥ 3. Johnson provides a finite presentation for H3. We extend his result to provide finite presentations of Hn for n> 3. We also establish exponential isoperimetric inequalities of Hn for n ≥ 3. 1. Intoduction One of major paradigms in geometric group theory is the idea that one can understand algebraic properties of groups by studying their actions on geometric spaces. There is a basic geometric object that one can associate to any finitely generated group: its Cayley graph with the word metric. Every pair (G,S) of a group and its finite generating set S has an associated word metric. The distance d(g, h) between elements g, h ∈ G is defined to be the length of the shortes