Hyperbolic Geometry and Distance Functions on Discrete Groups

TL;DR

This paper explores hyperbolic geometry, models, and metrics on discrete groups, proving key differences in metric equivalences for groups like PSL(2,Z) and PSL(n,Z) for n ≥ 3.

Contribution

It provides a detailed proof of the Lubotzky–Mozes–Raghunathan theorem regarding metric equivalences on discrete groups, highlighting distinctions between PSL(2,Z) and higher rank groups.

Findings

Word metric on PSL(2,Z) is not Lipschitz equivalent to the symmetric space metric.

For n ≥ 3, these metrics on PSL(n,Z) are Lipschitz equivalent.

The paper offers a comprehensive overview of hyperbolic models and group actions.

Abstract

Chapter 1 is a short history of non-Euclidean geometry, which synthesises my readings of mostly secondary sources. Chapter 2 presents each of the main models of hyperbolic geometry, and describes the tesselation of the upper half-plane induced by the action of $PSL(2,\mathbb{Z})$. Chapter 3 gives background on symmetric spaces and word metrics. Chapter 4 then contains a careful proof of the following theorem of Lubotzky--Mozes--Raghunathan: the word metric on $PSL(2,\mathbb{Z})$ is not Lipschitz equivalent to the metric induced by its action on the associated symmetric space (the upper half-plane), but for $n \geq 3$, these two metrics on $PSL(n,\mathbb{Z})$ are Lipschitz equivalent.