Strict dead end elements in free soluble groups

Abstract

Let $G$ be a group generated by a finite set $A$. An element $g\in G$ is a strict dead end of depth $k$ (with respect to $A$) if $|g|>|g{a}_1|>|g{a}_1{a}_2|>...>|g{a}_1{a}_2...a_k|$ for any $a_1,a_2,...,a_k\in A^{\pm 1}$ such that the word $a_1{a}_2...a_k$ is freely irreducible. (Here $|g|$ is the distance from $g$ to the identity in the Cayley graph of $G$.) We show that in finitely generated free soluble groups of degree $d\ge 2$ there exist strict dead elements of depth $k=k(d)$, which grows exponentially with respect to $d$.

11 pages