Ian J. Leary

On algebraic and geometric dimensions for groups with torsion

We argue that the geometric dimension of a discrete group G ought to be defined to be the minimal dimension of a model for the universal proper G-space rather than the minimal dimension of a model for the universal free G-space. For torsion-free groups, these two quantities are equal, but the new quantity can be finite for groups containing torsion whereas the old one cannot. There is an analogue of cohomological dimension (defined in terms of Bredon cohomology) for which analogues of the Eilenberg-Ganea and Stalling-Swan theorems (due to W. Lueck and M. J. Dunwoody respectively) hold. We show that some groups constructed by M. Bestvina and M. Davis provide counterexamples to the analogue of the Eilenberg-Ganea conjecture.

The mod-p cohomology rings of some p-groups

We compute the mod-p cohomology rings of a family of p-groups. This paper completes the computation of the mod-p cohomology rings of groups of order p^3.

Every CW-complex is a classifying space for proper bundles

Abstract We prove that, up to homotopy equivalence, every connected CW-complex is the quotient of a contractible complex by a proper action of a discrete group, and that every CW-complex is the quotient of an aspherical complex by an action of a group of order two.

The integral cohomology rings of some p-groups

We determine the integral cohomology rings of an infinite family of p -groups, for odd primes p , with cyclic derived subgroups. Our method involves embedding the groups in a compact Lie group of dimension one, and was suggested independently by P. H. Kropholler and J. Huebschmann. This construction has also been used by the author to calculate the mod- p cohomology of the same groups and by B. Moselle to obtain partial results concerning the mod- p cohomology of the extra special p -groups [7], [9].

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.

The cohomology of Bestvina–Brady groups

For each subcomplex of the standard CW-structure on any torus, we compute the homology of a certain infinite cyclic regular covering space. In all cases when the homology is finitely generated, we also compute the cohomology ring. For aspherical subcomplexes of the torus our computation gives the homology and cohomology of Bestvina-Brady groups. We compute the cohomological dimension of each of these groups over any field and over any subring of the rationals.

The spectrum of the Chern subring

Abstract. For certain subrings of the mod-p-cohomology of a compact Lie group, we give a description of the spectrum, analogous to Quillens description of the spectrum of the whole cohomology ring. Subrings to which our theorem applies include the Chern subring. Corollaries include a characterization of those groups for which the Chern subring is F-isomorphic to the cohomology ring.

The ` 2 -cohomology of hyperplane complements

We compute the` 2 -Betti numbers of the complement of a finite collection of affine hyperplanes in C n . At most one of the` 2 -Betti numbers is nonzero.

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 Bestvinas examples of groups whose vcd depends on the choice of ring.

On finite subgroups of groups of type VF

For any finite group Q not of prime power order, we construct a group G that is virtually of type F , contains infinitely many conjugacy classes of subgroups isomorphic to Q, and contains only finitely many conjugacy classes of other finite subgroups.