Dead end words in lamplighter groups and other wreath products
Sean Cleary, Jennifer Taback
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 Γ(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.
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