Some groups of finite homological type

For each n greater than or equal to zero we construct a torsion-free group that satisfies K. S. Brown's FHT condition and is type F(n), but is not of type FP(n+1). <br/

The L-two cohomology of Artin groups

For each Artin group we compute the reduced ℓ2-cohomology of its 'Salvetti complex'. This is a CW-complex which is conjectured to be a model for the classifying space of the Artin group. When this conjecture is known to hold our calculation describes the ℓ2-cohomology of the Artin group

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

A torsion projective class for a group algebra

We exhibit a cyclic-by-finite group G and two projective modules P and Q for the rational group algebra of G with the following properties: 1. P+P is isomorphic to Q+Q; 2. P is not stably isomorphic to Q; 3. after tensoring with the complex group algebra, P and Q become isomorphic. The proof that P and Q are not isomorphic is topological and involves the Mobius strip bundle over the circle

Some examples of discrete group actions on aspherical manifolds

We construct two classes of examples of virtually torsion-free groups G acting properly cocompactly on contractible manifolds X. In the first class of examples, the universal space for proper actions has no model with finitely many orbits of cells (and so the given manifold X cannot have this equivariant homotopy type). The reason is that the centralizers of some finite subgroups of G do not have finite-type classifying spaces. In the second class of examples, X is a CAT(0) manifold upon which G acts by isometries, and hence X is a model for the universal space for proper G actions. In these examples, the fixed-point sets for some finite subgroups of G are not manifolds and the centralizers of these subgroups are not virtual Poincare duality groups. <br/

Some remarks concerning degree zero complete cohomology

We describe degree zero mod-p complete cohomology modulo its radical in purely group-theoretic terms, for members of a class of groups that includes all groups of finite virtual cohomological dimension. We give examples to show that our description does not apply to all discrete groups. We also give examples of discrete groups to which Quillen's description of the ordinary mod-p cohomology ring (up to F-isomorphism) does not apply

On subgroups of Coxeter groups

The virtual cohomological dimension of a finitely generated Coxeter group G over a ring R is finite and known. We characterize the infinitely generated Coxeter groups of finite vcd, we give Coxeter groups that are virtual Poincare duality groups over some rings but not over others, and we exhibit a group whose vcd over the integers is three whereas its vcd over any field is two. We also give explicit presentations and Eilenberg-Mac Lane spaces for some of Bestvina's examples of groups whose vcd depends on the choice of ring

The spectrum of the Chern subring

For certain subrings of the mod-p cohomology ring of a compact Lie group, we give a description of the prime ideal spectrum, analogous to Quillen's description of the spectrum of the whole ring. Examples of such subrings include the Chern subring (the subring generated by Chern classes of all unitary representations), and for finite groups the subring generated by Chern classes of representations realizable over any specified field. As a corollary, we deduce that the inclusion of the Chern subring in the cohomology ring is an F-isomorphism for a compact Lie group G if and only if the following condition holds: For any homomorphism f between elementary abelian p-subgroups of G such that f(v) is always conjugate to v, there is an element g of G such that f is equal to conjugation by g

Artin HNN-extensions virtually embed in Artin groups

We define an Artin HNN-extension to be an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of any Artin HNN-extension embeds in an Artin group. Similarly, we show that every Coxeter HNN-extension virtually embeds in a Coxeter group

Presentations for subgroups of Artin groups

Recently, M. Bestvina and N. Brady have exhibited groups that are of type FP 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