Dead end words in lamplighter groups and other wreath products

Abstract

We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element $w$ in a group $G$ with finite generating set $X$ is a dead end element if no geodesic ray from the identity to $w$ in the Cayley graph $\Gamma(G,X)$ can be extended past $w$. Additionally, we describe some nonconvex behavior of paths between elements in these Cayley graphs and seesaw words, which are potential obstructions to these graphs satisfying the $k$-fellow traveller property.