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I. J. Leary

Publications and source records attributed to I. J. Leary.

6 recordsLinked to original sources
arXiv2009

Groups possessing extensive hierarchical decompositions

Kropholler's class of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever $G$ admits a finite-dimensional contractible $G$-CW-complex in which all stabilizer groups are in the class, then $G$ is itself in the class. Kropholler's class admits a hierarchical structure, i.e., a natural filtration indexed by the ordinals. For example, stage 0 of the hierarchy is the class of all finite groups, and stage 1 contains all groups of finite virtual cohomological dimension. We show that for each countable ordinal $α$, there is a countable group that is in Kropholler's class which does not appear until the $α+1$st stage of the hierarchy. Previously this was known only for $α= 0$, 1 and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in the third stage of the hierarchy.

math.GR↗
arXiv2009

Infinite groups with fixed point properties

We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in \mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group.

math.GR↗
arXiv2007

The Yagita invariant of general linear groups

We give a definition of the Yagita invariant at a prime p of an arbitrary group G, and compute the invariant for each prime for the general linear groups over any integrally closed subring of the complex numbers. We also compute the invariants for special linear groups over the same rings, except in some cases when both the degree of the linear group and the ring are `small' compared to p.

math.GR↗
arXiv2007

On universally stable elements

We show that certain subrings of the cohomology of a finite p-group P may be realised as the images of restriction from suitable virtually free groups. We deduce that the cohomology of P is a finite module for any such subring. Examples include the ring of `universally stable elements' defined by Evens and Priddy, and rings of invariants such as the mod-2 Dickson algebras.

math.GR↗
arXiv2007

On the GL(V)-module structure of K(n)^*(BV)

We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses) for the automorphism group of V. We use Brauer characters and computer calculations. We construct and implement an algorithm for finding permutation submodules of maximal dimension inside modules for p-groups in characteristic p.

math.AT↗