Dead ends on wreath products and lamplighter groups

TL;DR

This paper investigates the depth properties of lamplighter groups formed from finite and finitely generated groups, revealing conditions under which these groups have unbounded or bounded depth depending on their structure and generating sets.

Contribution

It generalizes previous results on lamplighter groups by characterizing unbounded depth for various group constructions and analyzing the geometric factors influencing this property.

Findings

Lamplighter groups over finite and finitely generated groups can have unbounded depth with suitable generators.

For abelian base groups, all standard generators lead to unbounded depth.

Lamplighter groups over free products of large finite cyclic groups can have bounded depth depending on the generating set.

Abstract

For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\wr B$ 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 $B=\mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ 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.