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

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.