Uncountably many quasi-isometric torsion-free groups
Vladimir Vankov
TL;DR
The work addresses the problem of producing uncountably many pairwise non-isomorphic torsion-free finitely generated groups that lie in a single quasi-isometry class. It deploys central extensions controlled by group cohomology, leveraging Schur coverings and the theory of bounded-valued (weakly bounded) cohomology to identify many quasi-isometric yet non-isomorphic quotients. By distinguishing a large set of cohomology classes in $H^2(G,\mathbb{Z})$ via the Kronecker pairing, the authors construct uncountably many central extensions $E_f$ of a carefully chosen torsion-free group $G$, all quasi-isometric to $G\times\mathbb{Z}$. The key technical advance is demonstrating the existence of uncountably many rich, weakly bounded, and pairwise distinguished cohomology classes, realized concretely using a torsion-free group with $H^2(G,\mathbb{Z})\cong\mathbb{Z}^{\mathbb{N}}$. This provides a broad method to generate large families of quasi-isometric, non-isomorphic groups with controlled torsion-free structure.
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.