Affiliation:
1. Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, F-75230 Paris Cedex 05, France
Abstract
For any finite group A and any finitely generated group B, we prove that the corresponding lamplighter group [Formula: see text] admits a standard generating set with unbounded depth, and that if B 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 H and K, 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 H and K. 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.
Publisher
World Scientific Pub Co Pte Ltd