Strict dead end elements in free soluble groups
  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 $|g|>|ga_1|>|ga_1a_2|>...>|ga_1a_2... a_k|$ for any $a_1,a_2, ..., a_k\in A^{\pm1}$ such that the word $a_1a_2... a_k$ 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 $d\ge2$ there exist strict dead elements of depth $k=k(d)$, which grows exponentially with respect to $d$. 
Victor Guba