[Paper Review] Conjugacy growth of finitely generated groups

Michael Hull, Denis Osin|arXiv (Cornell University)|Jul 10, 2011

Geometric and Algebraic Topology24 references3 citations

TL;DR

This paper completely characterizes which functions can arise as conjugacy growth functions of finitely generated groups: a function $ f: \mathbb{N} \to \mathbb{N} $ is realizable (up to equivalence) if and only if it is non-decreasing and bounded above by $ a^n $ for some $ a \geq 1 $. The authors construct a finitely generated group $ G $ with a subgroup $ H $ of index 2 such that $ H $ has only two conjugacy classes, while $ G $ has exponential conjugacy growth, proving that conjugacy growth is not a quasi-isometry invariant.

ABSTRACT

We show that every non-decreasing function $f\colon \mathbb N o \mathbb N$ bounded from above by $a^n$ for some $a\ge 1$ can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group $G$ and a subgroup $H\le G$ of index 2 such that $H$ has only 2 conjugacy classes while the conjugacy growth of $G$ is exponential. In particular, conjugacy growth is not a quasi-isometry invariant.

Motivation & Objective

• To determine which functions can arise as conjugacy growth functions of finitely generated groups.
• To investigate whether conjugacy growth is invariant under quasi-isometry, a central question in geometric group theory.
• To construct explicit examples demonstrating the extremal behavior of conjugacy growth, particularly in relation to subgroup structure.
• To establish a complete classification of conjugacy growth functions up to equivalence.

Proposed method

• The authors use small cancellation theory and HNN-extensions to inductively construct a group $ G $ with controlled conjugacy structure.
• They define a sequence of groups $ G(i) $, each satisfying specific properties: hyperbolicity relative to a cyclic subgroup $ C $, torsion-freeness, and controlled conjugacy behavior.
• The construction involves iteratively modifying groups via HNN-extensions and quotienting by normal subgroups to enforce conjugacy relations.
• The key technique is applying Theorem 6.2 (a small cancellation result) to ensure that certain elements become conjugate in the final quotient.
• They use the Svarč–Milnor lemma and word metrics to relate conjugacy growth to geometric properties of the group.
• The proof relies on inductive verification of structural properties (e.g., injectivity on $ C $, kernel control) across the group sequence.

Experimental results

Research questions

• RQ1 Which functions $ f: \mathbb{N} \to \mathbb{N} $ can be realized as the conjugacy growth function of a finitely generated group?
• RQ2 Is conjugacy growth invariant under quasi-isometry, given that ordinary growth is not?
• RQ3 Can a finitely generated group have exponential conjugacy growth while a finite-index subgroup has only finitely many conjugacy classes?
• RQ4 What structural conditions on a group ensure that its conjugacy growth is exponential?
• RQ5 How do hyperbolically embedded subgroups influence the conjugacy growth function?

Key findings

• A function $ f: \mathbb{N} \to \mathbb{N} $ is realizable as the conjugacy growth function of a finitely generated group if and only if it is non-decreasing and bounded above by $ a^n $ for some $ a \geq 1 $, up to the equivalence relation $ f \sim g $ iff $ f \preceq g $ and $ g \preceq f $.
• The authors construct a finitely generated group $ G $ with a subgroup $ H \leq G $ of index 2 such that $ H $ has exactly two conjugacy classes.
• Despite $ H $ having only two conjugacy classes, the conjugacy growth of $ G $ is exponential, demonstrating that conjugacy growth is not a quasi-isometry invariant.
• The group $ G $ is torsion-free and $ 2 $-generated, with the conjugacy growth function equivalent to $ 2^n $, confirming exponential growth.
• The construction relies on inductive HNN-extensions and small cancellation techniques to control conjugacy classes while preserving hyperbolicity relative to a cyclic subgroup.
• The result implies that conjugacy growth functions of subgroups in mapping class groups and other hyperbolic-like groups are also exponential, as shown by Theorem 1.1.