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Ged Corob Cook

Publications and source records attributed to Ged Corob Cook.

14 recordsLinked to original sources
arXiv2026

Solid Duality for Profinite Groups

We classify profinite groups that have a Poincar\'e-like duality between their homology and cohomology. Our proofs work over every profinite coefficient ring, and for profinite as well as discrete coefficients. We do not only unify and generalise existing results, but also construct two novel examples of duality groups. We prove our results via the framework of condensed mathematics. This recently developed setting is a natural home for profinite objects, and our work is among the first to leverage this for the study of the (co)homology of profinite groups. Establishing our duality results involves an investigation of homological finiteness properties in condensed mathematics.

math.GR↗
arXiv2022

Weil zeta functions of group representations over finite fields

In this article we define and study a zeta function $\zeta_G$ - similar to the Hasse-Weil zeta function - which enumerates absolutely irreducible representations over finite fields of a (profinite) group $G$. The zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value $\zeta_G(k)^{-1}$ at a positive integer $k$ coincides with the probability that $k$ random elements generate the completed group ring of $G$. The explicit formulas obtained so far suggest that $\zeta_G$ is rather well-behaved. A central object of this article is the abscissa of convergence $a(G)$ of $\zeta_G$. We calculate the abscissae for free abelian, free abelian pro-$p$, free pro-$p$, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the abscissae of free pro-$\mathfrak{C}$ groups, where $\mathfrak{C}$ is a class of finite groups with prescribed composition factors. We prove that every real number $a \geq 1$ is the abscissa $a(G)$ of some profinite group $G$. In addition, we show that the Euler factors of $\zeta_G$ are rational functions in $p^{-s}$ if $G$ is virtually abelian. For finite groups $G$ we calculate $\zeta_G$ using the rational representation theory of $G$.

math.GR↗
arXiv2021

Counting irreducible modules for profinite groups

This article is concerned with the representation growth of profinite groups over finite fields. We investigate the structure of groups with uniformly bounded exponential representation growth (UBERG). Using crown-based powers we obtain some necessary and some sufficient conditions for groups to have UBERG. As an application we prove that the class of UBERG groups is closed under split extensions but fails to be closed under extensions in general. On the other hand, we show that the closely related probabilistic finiteness property $PFP_1$ is closed under extensions. In addition, we prove that profinite groups of type $FP_1$ with UBERG are always finitely generated and we characterise UBERG in the class of pro-nilpotent groups. Using infinite products of finite groups, we construct several examples of profinite groups with unexpected properties: (1) an UBERG group which cannot be finitely generated, (2) a group of type $PFP_\infty$ which is not UBERG and not finitely generated and (3) a group of type $PFP_\infty$ with superexponential subgroup growth.

math.GR↗
arXiv2021

Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups

This article is part of the program of studying large-scale geometric properties of totally disconnected locally compact groups, TDLC-groups, by analogy with the theory for discrete groups. We provide a characterization of hyperbolic TDLC-groups, in terms of homological isoperimetric inequalities. This characterization is used to prove the main result of the article: for hyperbolic TDLC-groups with rational discrete cohomological dimension $\leq 2$, hyperbolicity is inherited by compactly presented closed subgroups. As a consequence, every compactly presented closed subgroup of the automorphism group $\mathrm{Aut}(X)$ of a negatively curved locally finite $2$-dimensional building $X$ is a hyperbolic TDLC-group, whenever $\mathrm{Aut}(X)$ acts with finitely many orbits on $X$. Examples where this result applies include hyperbolic Bourdon's buildings. We revisit the construction of small cancellation quotients of amalgamated free products, and verify that it provides examples of hyperbolic TDLC-groups of rational discrete cohomological dimension $2$ when applied to amalgamated products of profinite groups over open subgroups. We raise the question of whether our main result can be extended to locally compact hyperbolic groups if rational discrete cohomological dimension is replaced by asymptotic dimension. We prove that this is the case for discrete groups and sketch an argument for TDLC-groups.

math.GR↗
arXiv2020

Probabilistic finiteness properties for profinite groups

We introduce various probablistic finiteness conditions for profinite groups related to positive finite generation (PFG). We investigate completed group rings which are PFG as modules, and use this to answer a question of Kionke and the second author on positively finitely related groups. Using the theory of projective covers, we define and characterise a probabilistic version of the $\mathrm{FP}_n$ property for profinite groups, called $\mathrm{PFP}_n$. Finally, we prove how these conditions are related to previously defined finiteness conditions and each other.

math.GR↗
arXiv2019

A property of the lamplighter group

We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.

math.GR↗
arXiv2019

Finiteness properties of totally disconnected locally compact groups

In this paper we investigate finiteness properties of totally disconnected locally compact groups for general commutative rings $R$, in particular for $R = \mathbb{Z}$ and $R= \mathbb{Q}$. We show these properties satisfy many analogous results to the case of discrete groups, and we provide analogues of the famous Bieri's and Brown's criteria for finiteness properties and deduce that both $FP_n$-properties and $F_n$-properties are quasi-isometric invariant. Moreover, we introduce graph-wreath products in the category of totally disconnected locally compact groups and discuss their finiteness properties.

math.GR↗
arXiv2018

Homotopical Algebra in Categories with Enough Projectives

For a complete and cocomplete category $\mathcal{C}$ with a well-behaved class of `projectives' $\bar{\mathcal{P}}$, we construct a model structure on the category $s\mathcal{C}$ of simplicial objects in $\mathcal{C}$ where the weak equivalences, fibrations and cofibrations are defined in terms of $\bar{\mathcal{P}}$. This holds in particular when $\mathcal{C}$ is $\mathcal{U}$, the category of compactly generated, weakly Hausdorff spaces, and $\bar{\mathcal{P}}$ is the class of compact Hausdorff spaces. We also construct a new model structure on $\mathcal{U}$ itself, where the cofibrant spaces are generalisations of CW-complexes allowing spaces, rather than sets, of $n$-cells to be attached. The singular simplicial complex and geometric realisation functors give a Quillen adjunction between these model structures. For a space in $\mathcal{U}$, these structures allow the definition of homotopy group objects in the exact completion of $\mathcal{U}$, which are invariant under weak equivalence and have a lot of the nice properties usually expected of homotopy groups. There is a long exact sequence of homotopy group objects arising from a fibre sequence in $\mathcal{U}$. Working along similar lines, we study homological algebra in categories of internal modules in $\mathcal{U}$, getting in particular a Lyndon--Hochschild--Serre spectral sequence for extensions of topological groups in $\mathcal{U}$.

math.CT↗
arXiv2018

Eilenberg--Mac Lane Spaces for Topological Groups

The goal of this paper is to establish a topological version of the notion of an Eilenberg-Mac Lane space. If $X$ is a pointed topological space, $π_1(X)$ has a natural topology coming from the compact-open topology on the space of maps $S^1 \to X$. In general the construction does not produce a topological group because it is possible to create examples where the group multiplication $π_1(X) \times π_1(X) \to π_1(X)$ is discontinuous. This failure to obtain a topological group has been noticed by others, for example Fabel. However, if we work in the category of compactly generated, weakly Hausdorff spaces, we may retopologise both the space of maps $S^1 \to X$ and the product $π_1(X) \times π_1(X)$ with compactly generated topologies to get that $π_1(X)$ is a group object in this category. Such group objects are known as $k$-groups. Next we construct the Eilenberg-Mac Lane space $K(G,1)$ for any totally path-disconnected $k$-group $G$. The main point of this paper is to show that, for such a $G$, $π_1(K(G,1))$ is isomorphic to $G$ in the category of $k$-groups. All totally disconnected locally compact groups are $k$-groups and so our results apply in particular to profinite groups. This answers questions that have been raised by Sauer. We also show that there are Mayer-Vietoris sequences and a Seifert-van Kampen theorem in this theory. The theory requires a careful analysis using model structures and other homotopical structures on cartesian closed categories as we shall see that no theory can be comfortably developed in the classical world.

math.GR↗
arXiv2016

Continuous cohomology and homology of profinite groups

We develop cohomological and homological theories for a profinite group $G$ with coefficients in the Pontryagin dual categories of pro-discrete and ind-profinite $G$-modules, respectively. The standard results of group (co)homology hold for this theory: we prove versions of the Universal Coefficient Theorem, the Lyndon-Hochschild-Serre spectral sequence and Shapiro's Lemma.

math.GR↗
arXiv2015

Bieri-Eckmann Criteria for Profinite Groups

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type $\operatorname{FP}_n$ over a profinite ring $R$, analogous to the Bieri-Eckmann criteria for abstract groups. We use these to prove that the class of groups of type $\operatorname{FP}_n$ is closed under extensions, quotients by subgroups of type $\operatorname{FP}_n$, proper amalgamated free products and proper $\operatorname{HNN}$-extensions, for each $n$. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type $\operatorname{FP}_\infty$ over all profinite $R$. For any class $\mathcal{C}$ of finite groups closed under subgroups, quotients and extensions, we also construct pro-$\mathcal{C}$ groups of type $\operatorname{FP}_n$ but not of type $\operatorname{FP}_{n+1}$ over $\mathbb{Z}_{\hat{\mathcal{C}}}$ for each $n$. Finally, we show that the natural analogue of the usual condition measuring when pro-$p$ groups are of type $\operatorname{FP}_n$ fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.

math.GR↗
arXiv2014

On Profinite Groups of Type $\operatorname{FP}_\infty$

Suppose $R$ is a profinite ring. We construct a large class of profinite groups $\widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$, including all soluble profinite groups and profinite groups of finite cohomological dimension over $R$. We show that, if $G \in \widehat{{\scriptstyle\bf L}'{\scriptstyle\bf H}_R}\mathfrak{F}$ is of type $\operatorname{FP}_\infty$ over $R$, then there is some $n$ such that $H_R^n(G,R [[ G ]]) \neq 0$, and deduce that torsion-free soluble pro-$p$ groups of type $\operatorname{FP}_\infty$ over $\mathbb{Z}_p$ have finite rank, thus answering the torsion-free case of a conjecture of Kropholler.

math.GR↗
arXiv2014

A Note on Homology over Functor Categories

It is known that, for $C$ an abelian category and $I$ small, the functor category $C^I$ is again abelian; thus we can do homology in such categories, and examine how it relates to homology in $C$ itself. However, there does not seem to be any good reference collecting these ideas together. This article seeks to fill the gap by showing that homology in $C^I$ behaves as one would expect.

math.CT↗