The Conjugacy Ratio of Abelian-by-Cyclic Groups

2024/10/30 by Guo, David · 1 citation
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2410.23034

openalex publication_date 2024/10/30 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

Let G = K \rtimes ⟨ t ⟩ be a finitely generated group where K is abelian and ⟨ t⟩ is the infinite cyclic group. Let R be a finite symmetric subset of K such that S = \ (r,1),(0,t^{± 1}) | r ∈ R \ is a generating set of G. We prove that the spherical conjugacy ratio, and hence the conjugacy ratio, of G with respect to S is 0 unless G is virtually abelian, confirming a conjecture of Ciobanu, Cox and Martino in this case. We also show that the Baumslag--Solitar group BS(1,2) has a one-sided Følner sequence F such that the conjugacy ratio with respect to F is non-zero, even though BS(1,2) is not virtually abelian. This is in contrast to two-sided Følner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided Følner sequence is positive if and only if the group is virtually abelian.

Cited by

• Twisted commutativity and conjugacy ratio in groups

Related

• Free-by-cyclic groups have solvable conjugacy problem
• Twisted commutativity and conjugacy ratio in groups
• Multiplying a conjugacy class by its inverse in a finite group
• The conjugacy problem in free solvable groups and wreath product of abelian groups is in TC⁰
• Conjugacy depth function for generalised lamplighter groups