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Twisted commutativity and conjugacy ratio in groups

Laura Ciobanu, Gemma Crowe, Pieter Senden

TL;DR

This work introduces two density-type invariants in finitely generated groups: the degree of twisted commutativity $\mathrm{tdc}_X(\varphi,G)$ and the twisted conjugacy ratio $\mathrm{tcr}_X(\varphi,G)$, twisted by an endomorphism $\varphi$. It proves sharp results across growth regimes: $\mathrm{tdc}_X(\varphi,G)=0$ for all endomorphisms when $G$ is a free group, and for residually finite, non-virtually abelian groups with stable subexponential growth; in contrast, for infinite virtually abelian $G$, $\mathrm{tdc}_X(\varphi,G)>0$ iff there exists $g\in G$ with $(\iota_g\circ\varphi)|_A=\mathrm{Id}_A$, linking the twist to the action on a finite-index torsion-free abelian subgroup $A$. The twisted conjugacy ratio is shown to be positive under similar virtual-abelian criteria, while many exponential-growth examples (e.g., hyperbolic groups with finite-order automorphisms, certain Artin groups, and torus-knot groups) satisfy $\mathrm{tcr}_X(\varphi,G)=0$. The paper also establishes that $\mathrm{tcr}_X(\varphi,G)\le\mathrm{tdc}(\bar{\varphi},G/N)$ for finite-index quotients, tying the twisted growth behavior to quotient dynamics. Overall, these results extend the classical degree of commutativity and conjugacy ratio to twisted settings, provide concrete computations for key group classes, and raise open questions about generator-independence and broader applicability across group families.

Abstract

In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity $\mathrm{tdc}_X(\varphi, G)$ generalises the degree of commutativity of $G$, by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius $n$ of $G$, as $n \rightarrow \infty$, where the twisting is done with respect to an endomorphism $\varphi$ of $G$. We compute $\mathrm{tdc}_X(\varphi, G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio $\mathrm{tcr}_{X}(\varphi, G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute $\mathrm{tcr}_{X}(\varphi, G)$ for virtually abelian groups, and give examples of groups of exponential growth such that $\mathrm{tcr}_{X}(\varphi, G) = 0$.

Twisted commutativity and conjugacy ratio in groups

TL;DR

This work introduces two density-type invariants in finitely generated groups: the degree of twisted commutativity and the twisted conjugacy ratio , twisted by an endomorphism . It proves sharp results across growth regimes: for all endomorphisms when is a free group, and for residually finite, non-virtually abelian groups with stable subexponential growth; in contrast, for infinite virtually abelian , iff there exists with , linking the twist to the action on a finite-index torsion-free abelian subgroup . The twisted conjugacy ratio is shown to be positive under similar virtual-abelian criteria, while many exponential-growth examples (e.g., hyperbolic groups with finite-order automorphisms, certain Artin groups, and torus-knot groups) satisfy . The paper also establishes that for finite-index quotients, tying the twisted growth behavior to quotient dynamics. Overall, these results extend the classical degree of commutativity and conjugacy ratio to twisted settings, provide concrete computations for key group classes, and raise open questions about generator-independence and broader applicability across group families.

Abstract

In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group . The degree of twisted commutativity generalises the degree of commutativity of , by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius of , as , where the twisting is done with respect to an endomorphism of . We compute for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio , which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute for virtually abelian groups, and give examples of groups of exponential growth such that .
Paper Structure (13 sections, 29 theorems, 93 equations, 2 figures)

This paper contains 13 sections, 29 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Twisted conjugacy, twisted commutativity, and fixed subgroups
  4. Groups with unique roots
  5. Growth functions and growth rates
  6. Free groups
  7. Twisted commutativity and subexponential growth
  8. Non-virtually abelian groups of stable subexponential growth
  9. Virtually abelian groups
  10. The twisted conjugacy ratio in groups
  11. Groups of stable subexponential growth
  12. Groups of exponential growth
  13. Reflections and open questions

Key Result

Lemma 2.1

Let $G$ be a group, $\varphi \in \mathop{\mathrm{End}}\nolimits(G)$ and $g \in G$. Then $\mathop{\mathrm{Stab}}\nolimits_{\varphi}(g) = \mathop{\mathrm{Fix}}\nolimits(\iota_{g} \circ \varphi)$.

Figures (2)

  • Figure 1: Topological graph $G$ gives $u(C_H)$ (with $\circ$ in red) by the subdivision of $e$
  • Figure 2: The set $Top(2)$ consists of $3$ graphs

Theorems & Definitions (60)

  • Conjecture 1.1
  • Lemma 2.1: Senden23
  • Proposition 2.2
  • proof
  • Proposition 2.3: FelshtynHill94
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • ...and 50 more