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, ℓ2, and compactly supported cohomology of these groups. For each n⩾4 we construct a closed aspherical n‐manifold that admits an uncountable family of acyclic regular coverings with non‐isomorphic covering groups.
Various notions of dimension for discrete groups are compared. A group is exhibited that acts with finite stabilizers on an acyclic 2-complex in such a way that the fixed point subcomplex for any non-trivial finite subgroup is contractible, but such that the group does not admit any such action on a contractible 2-complex. This group affords a counterexample to a natural generalization of the Eilenberg-Ganea conjecture.
Abstract. Recently, M. Bestvina and N. Brady have exhibited groups that are of type F P but not finitely presented. We give explicit presentations for groups of the type considered by Bestvina-Brady. This leads to algebraic proofs of some of their results.
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