Uncountably many groups of type $FP$

TL;DR

This paper constructs uncountably many discrete groups of type $FP$, analyzes their cohomology, and provides geometric examples with uncountable families of acyclic coverings, advancing understanding of group and manifold structures.

Contribution

It introduces new uncountably many groups of type $FP$ that do not embed in finitely presented groups and computes their cohomology, also constructing novel geometric examples.

Findings

Uncountably many groups of type $FP$ constructed.

Computed various cohomology theories for these groups.

Provided geometric examples with uncountable acyclic coverings.

Abstract

We construct uncountably many discrete groups of type $FP$; in particular we construct groups of type $FP$ that do not embed in any finitely presented group. We compute the ordinary, $\ell^2$- and compactly-supported cohomology of these groups. For each $n\geq 4$ we construct a closed aspherical $n$-manifold that admits an uncountable family of acyclic regular coverings with non-isomorphic covering groups.