arXiv subjects

Ian J. Leary

Publications and source records attributed to Ian J. Leary.

At least 19 recordsLinked to original sources

arXiv2026

Universal Structure of Graph Product Kernels

Let $G_Γ$ be a graph product over a finite simplicial graph $Γ$, and let $K_Γ$ denote the kernel of the canonical homomorphism from $G_Γ$ to the direct product of its vertex groups. It is known that, up to isomorphism, $K_Γ$ depends only on the underlying graph $Γ$ and the cardinalities of the vertex groups. In this paper we establish a functorial refinement of this fact. We show that any collection of set maps between the vertex groups naturally induces a homomorphism between the corresponding kernels, and that this construction is functorial. Several applications are discussed.

math.GR↗

arXiv2025

Spectra of subrings of cohomology generated by characteristic classes for fusion systems

If $\mathcal{F}$ is a saturated fusion system on a finite $p$-group $S$, we define the Chern subring $Ch(\mathcal{F})$ of $\mathcal{F}$ to be the subring of the mod-$p$ cohomology $H^*(S)$ of $S$ generated by the Chern classes of $\mathcal{F}$-stable representations of $S$. We show that $Ch(\mathcal{F})$ is contained in $H^*(\mathcal{F})$ and apply a result of Green and the first author to describe its spectrum in terms of a certain category of elementary abelian subgroups of $S$. We obtain similar results for various related subrings, including those generated by characteristic classes of $\mathcal{F}$-stable $S$-sets.

math.GR↗

arXiv2023

Coherence for elementary amenable groups

We prove that for an elementary amenable group, coherence of the group, homological coherence of the group, and coherence of the integral group ring are all equivalent. This generalises a result of Bieri and Strebel for finitely generated soluble groups.

math.GR↗

arXiv2022

Homological growth of Artin kernels in positive characteristic

We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina--Brady groups. Specifically, we prove that the mod $p$ homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.

math.GR↗

arXiv2022

A note on torsion length and torsion subgroups

Answering Questions 19.23 and 19.24 from the Kourovka notebook we construct polycyclic groups with arbitrary torsion lengths and give examples of finitely presented groups whose quotients by their torsion subgroups are not finitely presented.

math.GR↗

arXiv2021

Commensurating HNN-extensions: non-positive curvature and biautomaticity

We show that the commensurator of any quasiconvex abelian subgroup in a biautomatic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this criterion we exhibit groups that are CAT(0) but not biautomatic. These groups also resolve a number of other questions concerning CAT(0) groups.

math.GR↗

arXiv2019

The Yagita invariant of symplectic groups of large rank

Fix a prime $p$, and let $R$ be any subring of the complex numbers that is either integrally closed or contains a primitive $p$th root of 1. For each $n\geq p-1$ we compute the Yagita invariant at the prime $p$ for the symplectic group $Sp(2n,R)$.

math.GR↗

arXiv2018

Uncountably many groups of type $FP$

We construct uncountably many discrete groups of type $FP$; in particular we construct groups of type $FP$ that do not embed in any finitely presented group. We compute the ordinary, $\ell^2$- and compactly-supported cohomology of these groups. For each $n\geq 4$ we construct a closed aspherical $n$-manifold that admits an uncountable family of acyclic regular coverings with non-isomorphic covering groups.

math.GR↗

arXiv2016

Equivariant vector bundles over classifying spaces for proper actions

Let $G$ be an infinite discrete group and let $\underline{E}G$ be a classifying space for proper actions of $G$. Every $G$-equivariant vector bundle over $\underline{E}G$ gives rise to a compatible collection of representations of the finite subgroups of $G$. We give the first examples of groups $G$ with a cocompact classifying space for proper actions $\underline{E}G$ admitting a compatible collection of representations of the finite subgroups of $G$ that does not come from a $G$-equivariant (virtual) vector bundle over $\underline{E}G$. This implies that the Atiyah-Hirzeburch spectral sequence computing the $G$-equivariant topological $K$-theory of $\underline{E}G$ has non-zero differentials. On the other hand, we show that for right angled Coxeter groups this spectral sequence always collapes at the second page and compute the $K$-theory of the classifying space of a right angled Coxeter group.

math.AT↗

arXiv2014

An Eilenberg-Ganea Phenomenon for Actions with Virtually Cyclic Stabilisers

In dimension 3 and above, Bredon cohomology gives an acurate purely algebraic description of the minimal dimension of the classifying space for actions of a group with stabilisers in any given family of subgroups. For some Coxeter groups and the family of virtually cyclic subgroups we show that the Bredon cohomological dimension is 2 while the Bredon geometric dimension is 3.

math.GR↗

arXiv2012

Presentations for subgroups of Artin groups

For a connected graph L, let G(L) be a group with generators the vertex set of L, subject only to the relations that the ends of each edge commute. Now let H(L) be the kernel of the homomorphism from G(L) to the integers that takes each vertex to 1. M. Bestvina and N. Brady have shown that finiteness properties of H(L) are intimately related to the topology of the clique complex of L. We give a presentation for H(L), with generators the edges of L, and an infinite family of relators for each 1-cycle in L. In the case when the clique complex for L is simply-connected, we give a finite presentation for H(L), with generators the edges (or 2-cliques) of L, and two relators for each 3-clique in L.

math.GR↗

arXiv2012

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.

math.GR↗

arXiv2012

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.

math.GR↗

arXiv2012

A metric Kan-Thurston theorem

For every simplicial complex X, we construct a locally CAT(0) cubical complex T_X, a cellular isometric involution i on T_X and a map t_X from T_X to X with the following properties: t_Xi = t_X; t_X is a homology isomorphism; the induced map from the quotient of T_X by the involution i to X is a homotopy equivalence; the induced map from the fixed point subspace for i in T_X to X is a homology isomorphism. The construction is functorial in X. One corollary is an equivariant Kan-Thurston theorem: every connected proper G-CW-complex has the same equivariant homology as the classifying space for proper actions of some other group. From this we obtain an extension of Quillen's theorem on the spectrum of an equivariant cohomology ring and an extension of a result of Block concerning assembly conjectures. Another corollary of our main result is that there can be no algorithm to decide whether a CAT(0) cubical group is generated by torsion. In appendices we prove some foundational results concerning cubical complexes, including the infinite dimensional case. We characterize the cubical complexes for which the natural metric is complete; we establish Gromov's criterion for infinite-dimensional cubical complexes; we show that every CAT(0) cube complex is cubical; we deduce that the second cubical subdivision of any locally CAT(0) cube complex is cubical.

math.GR↗

arXiv2009

On groups acting on contractible spaces with stabilizers of prime power order

We study actions of discrete groups on contractible topological spaces in which either (1) all stabilizers lie in the family of subgroups of prime power order or (2) all stabilizers lie in the family of finite subgroups. We compare the classifying spaces for actions with stabilizers in these two families, the Kropholler hierarchies build on these two families, and group cohomology relative to these two families.

math.GR↗

arXiv2008

Artin HNN-extensions virtually embed in Artin groups

An Artin HNN-extension is 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 an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.

math.GR↗

arXiv2007

A differential in the Lyndon-Hochschild-Serre spectral sequence

We consider the Lyndon-Hochschild-Serre spectral sequence with mod-p coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential. Using this result we deduce a similar formula for the Serre spectral sequence for a principal fibration with fibre the classifying space of a cyclic p-group. The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in the cohomology of a finite abelian group. As an example we determine the Poincare series for the mod-3 cohomology of various 3-groups. Remarks. 1) My definition of the higher differentials $d_i$ for $i\geq 2$ in the spectral sequence for a double chain complex differs from the usual one by a factor of $(-1)^{i+1}$. Both conventions are consistent, but the usual definition has the advantage of agreeing with the ``obvious'' definition of the differentials in the spectral sequence for the associated filtered chain complex. All of the theorems in this paper remain true exactly as stated if the more usual definition of $d_i$ is taken. 2) Carles Broto found a small mistake in this paper: the result for fibrations with fibre the classifying space of a cyclic group is stated for arbitrary fibrations, although it is only proved for principal fibrations. Since it is apparent from the first sentence of the proof that only principal fibrations are being considered, I have not bothered to publish an erratum.

math.AT↗

arXiv2007

p-Groups are not determined by their integral cohomology groups

For each prime p, we exhibit pairs of p-groups all of whose integral cohomology groups are isomorphic. The method used involves very little calculation. The groups are exhibited as kernels of homomorphisms from a compact Lie group G to U(1), and the main result is that kernels of `similar' elements of Hom(G,U(1)) have isomorphic integral cohomology groups. The 2-groups constructed in this version have been corrected (there was a mistake in the presentations given in the published paper).

math.AT↗