Continuously Many Quasi-isometry Classes of Residually Finite Groups

TL;DR

This paper demonstrates that by varying a subset of positive integers, one can generate a continuum of finitely generated residually finite small cancellation groups that are distinct up to quasi-isometry.

Contribution

It introduces a method to produce continuously many quasi-isometry classes of residually finite groups via small cancellation quotients of free groups.

Findings

Varying subset S yields infinitely many non-quasi-isometric groups.

All groups are finitely generated and residually finite.

Groups are constructed as quotients of F_2 depending on S.

Abstract

We study a family of finitely generated residually finite small cancellation groups. These groups are quotients of $F_2$ depending on a subset $S$ of positive integers. Varying $S$ yields continuously many groups up to quasi-isometry.