Warren Dicks

On semilocal rings

We give several characterizations of semilocal rings and deduce that rationally closed subrings of semisimple artinian rings are semilocal, that artinian modules have semilocal endomorphism rings, and that artinian modules cancel from direct sums.

The Group Fixed by a Family of Injective Endomorphisms of a Free Group

Groupoids Measuring devices Properties of the basic operations Minimal representatives and fixed subgroupoids Open problems Bibliography Index.

The Spectral Measure of Certain Elements of the Complex Group Ring of a Wreath Product

We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk and Zuk, and later used by them, together with Linnell and Schick, to produce a counterexample to a strong version of the Atiyah conjecture about the range of L2-Betti numbers. We use our results to construct manifolds with certain L2-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.

Groups, trees, and projective modules

Groups acting on graphs.- Fundamental groups.- Decompositions.- Cohomological dimension one.

Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture

SummaryWe show that Walter Neumanns strengthened form of Hanna Neumanns conjecture on the best possible upper bound for the rank of the intersection of two subgroups of a free group is equivalent to a conjecture on the best possible upper bound for the number of edges in a bipartite graph with a certain weak symmetry condition. We illustrate the usefulness of this equivalence by deriving relatively easily certain previously known results.

Hilbert series of fixed free algebras and noncommutative classical invariant theory

We compute an asymptotic formula for the number of invariants of a given degree, which compares nicely with the corresponding result in the classical commutative case. In Section 6 we show that if G is an infinite cyclic group generated by a unipotent matrix then

On the intersection of free subgroups in free products of groups

Let (Gi j i 2 I) be a family of groups, let F be a free group, and let G = F ⁄ ⁄ i2I Gi, the free product of F and all the Gi.

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 boundary of the Gieseking tree in hyperbolic three-space

Abstract We give an elementary proof of the Cannon–Thurston Theorem in the case of the Gieseking manifold. We do not use Thurstons structure theory for Kleinian groups but simply calculate with two-by-two complex matrices. We work entirely on the boundary, using ends of trees, and obtain pictures of the regions which are successively filled in by the Peano curve of Cannon and Thurston.

L2-Betti numbers of one-relator groups

We determine the L2-Betti numbers of all one-relator groups and all surface-plus-one-relation groups. We also obtain some information about the L2-cohomology of left-orderable groups, and deduce the non-L2 result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free.