21 research outputs found
On profinite groups of type FP infinity
No full textSuppose R is a profinite ring. We construct a large class of profinite groups View the MathML source, including all soluble profinite groups and profinite groups of finite cohomological dimension over R . We show that, if View the MathML source is of type FP? over R, then there is some n such that View the MathML source, and deduce that torsion-free soluble pro-p groups of type FP? over Zp have finite rank, thus answering the torsion-free case of a conjecture of Kropholler
Bieri-Eckmann criteria for profinite groups
No full textIn this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type 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 FP n is closed under extensions, quotients by subgroups of type FP n , proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP? over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FP n but not of type FP n+1 over Z ? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler
Continuous cohomology and homology of profinite groups
No full textWe 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
A property of the lamplighter group
Get PDFWe 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
On the dimension of classifying spaces for families of abelian subgroups
No full textWe show that a finitely generated abelian group G of torsion free rank n ≥ 1 admits a n + r dimensional model for EFrG, where Fr is the family of subgroups of torsion-free rank less than or equal to r ≥ 0
Corrigendum to “Finiteness properties of totally disconnected locally compact groups” [J. Algebra 543 (2020) 54–97]
No full textCastellano I, Cook GC. Corrigendum to “Finiteness properties of totally disconnected locally compact groups” [J. Algebra 543 (2020) 54–97]. Journal of Algebra. 2024;647:906-909
Cohomology of Profinite Groups
No full textThe aim of this thesis is to study profinite groups of type FPn. These are groups G which admit a projective resolution P of Zˆ as a ZˆJGK-module such that P0, . .. , Pn are finitely generated, so this property can be studied using the tools of profinite group cohomology. In studying profinite groups it is often useful to consider their cohomology groups with profinite coefficients, but pre-existing theories of profinite cohomology do not allow profinite coefficients in sufficient generality for our purposes. Therefore we develop a new framework in which to study the homology and cohomology of profinite groups, which allows second countable profinite coefficients for all profinite groups. We prove that many of the results of abstract group cohomology hold here, including Shapiro’s Lemma, the Universal Coefficient Theorem and the Lyndon-Hochschild-Serre spectral sequence. We then use these homology and cohomology theories to study how being of type FPn controls the structure of a profinite group, and vice versa. We show for all n that the class of groups of type FPn is closed under extensions, quotients by subgroups of type FPn, proper amalgamated free products and proper HNNextensions, and hence that elementary amenable profinite groups of finite rank are of type FP∞. We construct profinite groups of type FPn but not FPn+1 for all n. Finally, we develop the theory of signed profinite permutation modules, and use these as coefficients for group cohomology to show that torsion-free soluble pro-p groups of type FP∞ have finite rank.<br/
Eilenberg–Mac Lane Spaces for Topological Groups
No full textIn this paper, we 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 → 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 ) × π 1 ( X ) → π 1 ( X ) is discontinuous. This discontinuity 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 → X and the product π 1 ( X ) × π 1 ( X ) with compactly generated topologies to see 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, answering a question of Sauer’s. We also show that analogues of the Mayer–Vietoris sequence and Seifert–van Kampen theorem hold in this context. 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
UROS Origami project
No full textThis is for data produced by my Origami project with Ewan Dalgliesh; initially, diagrams for each of the 2-fold axioms defined by Alperin and Lang. If more data is produced later, it will be added.</p
