Dead ends on wreath products and lamplighter groups

Abstract

For any finite group [Formula: see text] and any finitely generated group [Formula: see text], we prove that the corresponding lamplighter group [Formula: see text] admits a standard generating set with unbounded depth, and that if [Formula: see text] is abelian then the above is true for every standard generating set. This generalizes the case where [Formula: see text] together with its cyclic generator [S. Cleary and J. Taback, Dead end words in lamplighter groups and other wreath products, Q. J. Math. 56(2) (2005) 165–178]. When [Formula: see text] is the free product of two finite groups [Formula: see text] and [Formula: see text], we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of [Formula: see text] and [Formula: see text]. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.