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Automorphism growth and group decompositions

Elia Fioravanti

Abstract

Let $G$ be a finitely generated group with an automorphism $\varphi\in{\rm Aut}(G)$, or an outer automorphism $φ\in{\rm Out}(G)$. Suppose that $G$ decomposes into simpler pieces on which the growth behaviour of $\varphi$ and $φ$ is known, particularly as a direct product, free product, or graph of groups. This article is devoted to the (often not entirely straightforward) problem of deducing information about the growth rates of $\varphi$ and $φ$ on the whole $G$.

Automorphism growth and group decompositions

Abstract

Let be a finitely generated group with an automorphism , or an outer automorphism . Suppose that decomposes into simpler pieces on which the growth behaviour of and is known, particularly as a direct product, free product, or graph of groups. This article is devoted to the (often not entirely straightforward) problem of deducing information about the growth rates of and on the whole .
Paper Structure (9 sections, 12 theorems, 34 equations)

This paper contains 9 sections, 12 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Growth rates
  3. General properties
  4. Tameness and docility
  5. Direct products
  6. Abelian factors
  7. The general case
  8. Invariant graphs of groups
  9. Free products

Key Result

Lemma 2.3

The map $\varphi\colon G\rightarrow G$ is bi-Lipschitz with respect to both $|\cdot|$ and $\|\cdot\|$.

Theorems & Definitions (32)

  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Example 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 22 more