Strict dead end elements in free soluble groups

2005/08/22 by Victor Guba, Guba, Victor
Mathematics · #05C25 #20F32 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR #msc:05C25 #msc:20F32

paper · pdf · doi:10.48550/arxiv.math/0508422

11 pages

arxiv created 2005/08/22 · openalex publication_date 2005/08/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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|>|ga₁|>|ga₁a₂|>...>|ga₁a₂... a_k| for any a₁,a₂, ..., a_k∈ A^±1 such that the word a₁a₂... 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.

Related

• Some remarks on depth of dead ends in groups
• Arbitrarily large finite quotients imply no uniform bound on depth
• The asymptotic density of dead ends in non-amenable groups
• On k-free-like groups
• Finite soluble groups satisfying the replacement property