Some remarks on depth of dead ends in groups

Abstract

It is known, that the existence of dead ends (of arbitrary depth) in the Cayley graph of a group depends on the chosen set of generators. Nevertheless there exist many groups, which do not have dead ends of arbitrary depth with respect to any set of generators. Partial results in this direction were obtained by \v{S}uni\'c and by Warshall. We improve these results by showing that abelian groups only have finitely many dead ends and that groups with more than one end (in the sense of Hopf and Freudenthal) have only dead ends of bounded depth. Only few examples of groups with unbounded dead end depth are known. We show that the Houghton group \Hou with respect to a standard generating set is a further example. In addition we introduce a stronger notion of depth of a dead end, called strong depth. The Houghton group \Hou has unbounded strong depth with respect to the same standard generating set.