We calculate the centralizers of elements, finite subgroups and virtually cyclic subgroups of Houghton’s group Hn. We discuss various Bredon (co)homological finiteness conditions satisfied by Hn including the Bredon (co)homological dimension and FPn conditions, which are analogues of the ordinary cohomological dimension and FPn conditions, respectively.

Let S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.

In a soluble group G, if the metabelian subgroups have finite (or polycyclic, or Černikov) quotients by their centres, then so too does G.

Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincaré duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which $K_0$ contains torsion, and construct non-free stably free modules for the group algebras of certain virtually free groups.

AMS 2000 Mathematics subject classification: Primary 19A31. Secondary 16S34; 20J05; 55N15; 57P10