Shared Rough and Quasi-Isometries of Groups

TL;DR

This paper introduces a variation of quasi-isometry with uniform parameters to better understand group commensurability, comparing it with isometry and rough isometry approaches.

Contribution

It proposes a new geometric notion called shared rough and quasi-isometries, extending the concept of quasi-isometry with uniform parameters for families of generating systems.

Findings

Defines a new variation of quasi-isometry with uniform parameters

Establishes connections between quasi-isometry, rough isometry, and isometry

Provides a framework for understanding group commensurability geometrically

Abstract

We present a variation of quasi-isometry to approach the problem of defining a geometric notion equivalent to commensurability. In short, this variation can be summarized as "quasi-isometry with uniform parameters for a large enough family of generating systems". Two similar notions (using isometries and rough isometries instead, respectively) are presented alongside. This article is based mainly on a chapter of the author's doctoral thesis (\cite{Lochmann_dissertation}).