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 ≥4. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each n ≥4 there are uncountably many quasi-isometry classes of acyclic n -manifolds admitting free cocompact properly discontinuous discrete group actions.