Uncountably many quasi-isometry classes of groups of type $FP$

TL;DR

This paper demonstrates that there are uncountably many quasi-isometry classes of groups of type $FP$ and of acyclic $n$-manifolds, extending previous constructions to a broader classification.

Contribution

It proves that the previously constructed groups of type $FP$ and Poincaré duality groups form uncountably many quasi-isometry classes, revealing a richer geometric diversity.

Findings

Uncountably many quasi-isometry classes of groups of type $FP$.

Uncountably many quasi-isometry classes of acyclic $n$-manifolds.

Extension of previous results to broader geometric classifications.

Abstract

Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincar\'e duality groups for each $n\geq 4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions.