Strict Dead-End Elements in Free Soluble Groups

Abstract

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.

Key Words:

• Cayley graphs
• Dead end elements
• Free soluble groups

2000 Mathematics Subject Classification:

• Primary 20F32
• Secondary 05C25

ACKNOWLEDGMENT

This research is partially supported by the RFFI grant 05–01–00895 and the INTAS grant 99–1224.

Notes

Communicated by A. Yu. Olshanskii.