Some free-by-cyclic groups

We exhibit free-by-cyclic groups containing non-free locally-free subgroups, including some word hyperbolic examples. We also show that these groups are not subgroup separable. We use Bestvina-Brady Morse theory in our arguments

Property A and CAT(0) cube complexes

Property A is a non-equivariant analogue of amenability defined for metric spaces. Euclidean spaces and trees are examples of spaces with Property A. Simultaneously generalising these facts, we show that finite-dimensional CAT(0) cube complexes have Property A. We do not assume that the complex is locally finite. We also prove that given a discrete group acting properly on a finite-dimensional CAT(0) cube complex the stabilisers of vertices at infinity are amenable

Engulfing in word-hyperbolic groups

We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups

The singularity obstruction for group splittings

We study an obstruction to splitting a finitely generated group G as an amalgamated free product or HNN extension over a given subgroup H and show that when the obstruction is "small" G splits over a related subgroup. Applications are given which generalise decomposition theorems from low dimensional topology

Finding splittings of groups and three-manifolds

The class of almost finitely presented groups which admit a finite index subgroup which is an HNN extension is characterised using Scott's end invariant for a pair of groups. As in Scott's original paper on the subject the profinite topology plays a role

The subgroup separability of some amalgamated free products

SIGLEAvailable from British Library Document Supply Centre- DSC:DX95180 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

K-theory for subspaces of groups

We show that a subspace of a group carries a natural partial translation structure which gives rise to a C*-algebra similar to that of a reduced C*-algebra of a group. We investigate the question: Does the inclusion of a subspace into an ambient group induce a homomorphism of C*-algebras? We prove that the answer is affirmative for subspaces of groups with non-coarsely dense complement. We show that under this condition there exists an exact sequence of C*-algebras which is analogous to the Pimsner-Voiculescu extension

The geometry of cube complexes and the complexity of their fundamental groups

We investigate the geometry of geodesics in CAT(0) cube complexes. A group which acts cocompactly and properly discontinuously on such a complex is shown to have a biautomatic structure. There is a family of natural subgroups each of which is shown to be rational