Characterising quasi-isometries of the free group

TL;DR

This paper characterizes all self quasi-isometries of free groups by introducing mixed subtree quasi-isometries, showing they approximate any such quasi-isometry, and providing a construction method.

Contribution

It introduces mixed subtree quasi-isometries and proves they approximate all self quasi-isometries of free groups, offering a new structural understanding.

Findings

Any self quasi-isometry of a regular tree is close to a mixed-subtree quasi-isometry.

Provides a method to construct quasi-isometries of the free group.

Characterizes all self quasi-isometries of free groups via tree quasi-isometries.

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.