Characterising quasi-isometries of the free group

Goldsborough, Antoine; Zbinden, Stefanie

doi:10.2140/agt.2024.24.5211

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Characterising quasi-isometries of the free group

Antoine Goldsborough and Stefanie Zbinden

Algebraic & Geometric Topology 24 (2024) 5211–5219

DOI: 10.2140/agt.2024.24.5211

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.

Keywords

quasi-isometry, free group, geometric group theory

Mathematical Subject Classification

Primary: 20F65

References

Publication

Received: 18 August 2023

Revised: 12 December 2023

Accepted: 21 December 2023

Published: 27 December 2024

Authors

Antoine Goldsborough
Maxwell Institute and Department of Mathematics Heriot-Watt University Edinburgh United Kingdom

Stefanie Zbinden
Maxwell Institute and Department of Mathematics Heriot-Watt University Edinburgh United Kingdom

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Volume 24, issue 9 (2024)