Uncountably many quasi-isometry classes of groups of type FP

@article{Kropholler2020UncountablyMQ,
  title={Uncountably many quasi-isometry classes of groups of type FP},
  author={Robert P. Kropholler and Ian J. Leary and Ignat Soroko},
  journal={American Journal of Mathematics},
  year={2020},
  volume={142},
  pages={1931 - 1944}
}
Abstract:In an earlier paper, 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 show that those 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. 
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