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The twisted conjugacy growth of virtually abelian groups

Karel Dekimpe, Maarten Lathouwers

Abstract

In this paper, we study the asymptotics of several growth functions related to twisted conjugacy on virtually abelian groups. First, we study the twisted conjugacy growth function, which counts the number of twisted conjugacy classes intersecting the ball of radius r around the identity element. Thereafter we study the function that measures the size of the intersection of a given twisted conjugacy class with the balls around the identity element. Finally, we study the number of induced twisted conjugacy classes in certain finite quotients of the given virtually abelian group. In each of these cases we obtain a polynomial asymptotic behaviour of these growth functions.

The twisted conjugacy growth of virtually abelian groups

Abstract

In this paper, we study the asymptotics of several growth functions related to twisted conjugacy on virtually abelian groups. First, we study the twisted conjugacy growth function, which counts the number of twisted conjugacy classes intersecting the ball of radius r around the identity element. Thereafter we study the function that measures the size of the intersection of a given twisted conjugacy class with the balls around the identity element. Finally, we study the number of induced twisted conjugacy classes in certain finite quotients of the given virtually abelian group. In each of these cases we obtain a polynomial asymptotic behaviour of these growth functions.
Paper Structure (8 sections, 23 theorems, 119 equations)

This paper contains 8 sections, 23 theorems, 119 equations.

Table of Contents

  1. Twisted conjugacy growth
  2. Definitions
  3. Comparing asymptotic growth
  4. Twisted conjugacy growth of Z^n
  5. Twisted conjugacy growth of virtually abelian groups
  6. Twisted conjugacy class growth
  7. Twisted conjugacy quotient growth
  8. Examples

Key Result

Theorem A

Let $S$ be any finite generating set of ${\wiskunde Z}^n$ and $\varphi\in \mathop{\mathrm{End}}\nolimits({\wiskunde Z}^n)$, then

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • ...and 37 more