Strict Dead-End Elements in Free Soluble Groups

Let G be a group generated by a finite set A. An element g ∈ G is a strict dead end of depth k (with respect to A) if |g|>|ga ₁|>|ga ₁ a ₂|>···>|ga ₁ a ₂… a _k | for any a ₁, a ₂,…, a _k  ∈ A ^±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 ≥ 2 there exist strict dead elements of depth k = k(d), which grows exponentially with respect to d.