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Characterising quasi-isometries of the free group

Antoine Goldsborough, Stefanie Zbinden

TL;DR

The paper tackles the problem of characterizing self-quasi-isometries of the free group by focusing on regular trees, which are quasi-isometric to the free group. It introduces $D$-mixed subtree quasi-isometries as inductively defined maps on regular trees and proves that every self-quasi-isometry is at bounded distance from such a map, providing a concrete framework to describe and construct quasi-isometries of the free group $\,\mathbb{F}_2\$. This leads to a tangible understanding of $QI(\mathbb{F}_2)$ and offers new tools for constructing quasi-isometries with prescribed properties, such as altering random walk drift. The results have potential implications for broader questions about the quasi-isometry groups of free groups and related geometric structures.

Abstract

We introduce the notion of mixed subtree quasi-isometries, which are self quasi-isometries of regular trees built in a specific inductive way. We then show that any self quasi-isometry of a regular tree is at bounded distance from a mixed-subtree quasi-isometry. Since the free group is quasi-isometric to a regular tree, this provides a way to describe all self quasi-isometries of the free group. In doing this, we also give a way of constructing quasi-isometries of the free group.

Characterising quasi-isometries of the free group

TL;DR

The paper tackles the problem of characterizing self-quasi-isometries of the free group by focusing on regular trees, which are quasi-isometric to the free group. It introduces -mixed subtree quasi-isometries as inductively defined maps on regular trees and proves that every self-quasi-isometry is at bounded distance from such a map, providing a concrete framework to describe and construct quasi-isometries of the free group . This leads to a tangible understanding of and offers new tools for constructing quasi-isometries with prescribed properties, such as altering random walk drift. The results have potential implications for broader questions about the quasi-isometry groups of free groups and related geometric structures.

Abstract

We introduce the notion of mixed subtree quasi-isometries, which are self quasi-isometries of regular trees built in a specific inductive way. We then show that any self quasi-isometry of a regular tree is at bounded distance from a mixed-subtree quasi-isometry. Since the free group is quasi-isometric to a regular tree, this provides a way to describe all self quasi-isometries of the free group. In doing this, we also give a way of constructing quasi-isometries of the free group.
Paper Structure (5 sections, 6 theorems, 2 equations, 4 figures)

This paper contains 5 sections, 6 theorems, 2 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation on trees
  4. Properties of quasi-isometries of trees
  5. Quasi-isometries of regular trees

Key Result

Theorem 1.1

Let $T$ be a regular tree of degree at least 3, rooted at $v_0$. Let $f: T \to T$ be a $C$-quasi-isometry such that $f(v_0)=v_0$. Then there is a constant $D$ only depending on $C$ and a $D$-deep mixed subtree quasi-isometry $g: T\to T$ such that $f$ and $g$ are at bounded distance from each other.

Figures (4)

  • Figure 1: Images of geodesics coarsely surject onto the geodesic.
  • Figure 2: Quasi-isometries are at bounded distance from order-preserving quasi-isometries.
  • Figure 3: Illustration of the proof of Lemma \ref{['lemma:same_dist_properties']}.
  • Figure 4: Definition of $f'$

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 11 more