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Conjugacy Class Growth in Virtually Abelian Groups

Aram Dermenjian, Alex Evetts

TL;DR

The paper proves that conjugacy-class growth in finitely generated virtually abelian groups is always polynomial and provides a sharp description in affine Coxeter groups: the growth of a conjugacy class is asymptotically equivalent to n^{ell_R(u)} where u is the finite-part of the element w = t u in W ≅ T ⋊ W0. The approach uses a reduction to a normal abelian subgroup and an analysis of commutator subgroups to establish polynomial growth, while for affine Coxeter groups it leverages the translation–finite part decomposition, move-sets, and the notion of reflection length to relate growth degree to ell_R(u). The work clarifies the connection between algebraic structure (virtually abelian, reflection length) and geometric growth, and provides explicit degrees of growth in the affine Coxeter setting. This advances understanding of conjugacy dynamics in large-scale geometry contexts and yields computable invariants for growth in a broad class of groups.

Abstract

We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius $n$ in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class.

Conjugacy Class Growth in Virtually Abelian Groups

TL;DR

The paper proves that conjugacy-class growth in finitely generated virtually abelian groups is always polynomial and provides a sharp description in affine Coxeter groups: the growth of a conjugacy class is asymptotically equivalent to n^{ell_R(u)} where u is the finite-part of the element w = t u in W ≅ T ⋊ W0. The approach uses a reduction to a normal abelian subgroup and an analysis of commutator subgroups to establish polynomial growth, while for affine Coxeter groups it leverages the translation–finite part decomposition, move-sets, and the notion of reflection length to relate growth degree to ell_R(u). The work clarifies the connection between algebraic structure (virtually abelian, reflection length) and geometric growth, and provides explicit degrees of growth in the affine Coxeter setting. This advances understanding of conjugacy dynamics in large-scale geometry contexts and yields computable invariants for growth in a broad class of groups.

Abstract

We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class.
Paper Structure (11 sections, 18 theorems, 31 equations)

This paper contains 11 sections, 18 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic results
  3. Growth in groups
  4. Virtually abelian groups
  5. Virtually abelian groups
  6. Affine Coxeter groups
  7. Reflection length
  8. Affine Coxeter groups
  9. Movement
  10. Dimensions
  11. Conjugacy class growth

Key Result

Lemma 2.2

Fix any element $h\in G$. Then $\lvert\left\{g\in G\;\colon\;d_{G,S}(h,g)\leq n\right\}\rvert=\beta_{G}(n)$.

Theorems & Definitions (37)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Corollary 2.6
  • Remark 3.1
  • Proposition 3.2
  • ...and 27 more