Uncountably many quasi-isometric torsion-free groups

TL;DR

This paper constructs uncountably many torsion-free groups that are all quasi-isometric, using advanced algebraic and geometric tools like Schur covers and bounded cohomology, revealing rich diversity within a single quasi-isometry class.

Contribution

It introduces a novel method to generate uncountably many non-isomorphic torsion-free groups within the same quasi-isometry class using cohomological techniques.

Findings

Uncountably many non-isomorphic torsion-free groups are quasi-isometric.

The construction employs Schur covering groups and group cohomology.

Bounded-valued cohomology provides the geometric foundation for the results.

Abstract

We construct uncountably many finitely generated, pairwise non-isomorphic torsion-free groups, all of which fall into the same quasi-isometry class. This is done by considering Schur covering groups and group cohomology, with the necessary geometric ingredient coming from the theory of bounded-valued cohomology.