scholarly journals Decomposing locally compact groups into simple pieces

2010 ◽  
Vol 150 (1) ◽  
pp. 97-128 ◽  
Author(s):  
PIERRE-EMMANUEL CAPRACE ◽  
NICOLAS MONOD
Keyword(s):  
Simple Groups ◽  
Structure Theory ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Cocompact Subgroup ◽  
Simple Quotient ◽  
Compactly Generated

AbstractWe present a contribution to the structure theory of locally compact groups. The emphasis is on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple.Two appendices introduce results and examples around the concept of quasi-product.

scholarly journals LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART II: COMPACTLY GENERATED SIMPLE GROUPS

2017 ◽  
Vol 5 ◽  
Author(s):  
PIERRE-EMMANUEL CAPRACE ◽  
COLIN D. REID ◽  
GEORGE A. WILLIS
Keyword(s):  
Boolean Algebras ◽  
Simple Groups ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Composition Factors ◽  
Totally Disconnected ◽  
Compactly Generated ◽  
Centralizer Lattice

We use the structure lattice, introduced in Part I, to undertake a systematic study of the class $\mathscr{S}$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are nondiscrete. Given $G\in \mathscr{S}$, we show that compact open subgroups of $G$ involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centralizer lattice and local decomposition lattice of $G$ are Boolean algebras. We show that the $G$-action on the Stone space of those Boolean algebras is minimal, strongly proximal, and microsupported. Building upon those results, we obtain partial answers to the following key problems: Are all groups in $\mathscr{S}$ abstractly simple? Can a group in $\mathscr{S}$ be amenable? Can a group in $\mathscr{S}$ be such that the contraction groups of all of its elements are trivial?

scholarly journals LOCALLY COMPACT WREATH PRODUCTS

2018 ◽  
Vol 107 (1) ◽  
pp. 26-52 ◽  
Author(s):  
YVES CORNULIER
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Natural Extension ◽  
Wreath Products ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Intermediate Growth ◽  
Compactly Generated

Wreath products of nondiscrete locally compact groups are usually not locally compact groups, nor even topological groups. As a substitute introduce a natural extension of the wreath product construction to the setting of locally compact groups. Applying this construction, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without any open compact normal subgroup.

Locally compact groups: maximal compact subgroups and N-groups

1988 ◽  
Vol 104 (1) ◽  
pp. 47-64
Author(s):  
R. W. Bagley ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Compact Set ◽  
Locally Compact ◽  
Group A ◽  
Totally Disconnected ◽  
Compactly Generated

AbstractIf G is a locally compact group such thatG/G0contains a uniform compactly generated nilpotent subgroup, thenGhas a maximal compact normal subgroupKsuch thatG/Gis a Lie group. A topological groupGis an N-group if, for each neighbourhoodUof the identity and each compact setC⊂G, there is a neighbourhoodVof the identity such thatfor eachg∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

scholarly journals Totally disconnected, nilpotent, locally compact groups

1997 ◽  
Vol 55 (1) ◽  
pp. 143-146 ◽  
Author(s):  
G. Willis
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Totally Disconnected ◽  
Compactly Generated

It is shown that, if G is a totally disconnected, compactly generated and nilpotent locally compact group, then it has a base of neighbourhoods of the identity consisting of compact, open, normal subgroups. An example is given showing that the hypothesis that G be compactly generated is necessary.

scholarly journals Rational discrete first degree cohomology for totally disconnected locally compact groups

2018 ◽  
Vol 168 (2) ◽  
pp. 361-377 ◽  
Author(s):  
ILARIA CASTELLANO
Keyword(s):  
Cohomological Dimension ◽  
Finite Graph ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Fundamental Groups ◽  
Profinite Groups ◽  
Locally Compact ◽  
Number Of Ends ◽  
Totally Disconnected ◽  
Compactly Generated

AbstractIt is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information about the number of ends of G in the sense of H. Abels can be provided by dH1(G, Bi(G)), where Bi(G) is the rational discrete standard bimodule of G, and dH•(G, _) denotes rational discrete cohomology as introduced in [6].As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).

Quasi-actions and generalised Cayley–Abels graphs of locally compact groups

2015 ◽  
Vol 18 (1) ◽  
pp. 45-60
Author(s):  
Pekka Salmi
Keyword(s):  
Compact Group ◽  
Cayley Graph ◽  
Polynomial Growth ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Cocompact Lattices ◽  
Totally Disconnected ◽  
Compactly Generated

Abstract We define the notion of generalised Cayley–Abels graph for compactly generated locally compact groups in terms of quasi-actions. This extends the notion of Cayley–Abels graph of a compactly generated totally disconnected locally compact group, studied in particular by Krön and Möller under the name of rough Cayley graph (and relative Cayley graph). We construct a generalised Cayley–Abels graph for any compactly generated locally compact group using quasi-lattices and show uniqueness up to quasi-isometry. A class of examples is given by the Cayley graphs of cocompact lattices in compactly generated groups. As an application, we show that a compactly generated group has polynomial growth if and only if its generalised Cayley–Abels graph has polynomial growth (same for intermediate and exponential growth). Moreover, a unimodular compactly generated group is amenable if and only if its generalised Cayley–Abels graph is amenable as a metric space.

scholarly journals Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 190
Author(s):  
Karl H. Hofmann ◽  
Sidney A. Morris
Keyword(s):  
Vector Space ◽  
Lie Groups ◽  
Compact Subgroup ◽  
Natural Duality ◽  
Structure Theory ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Quarter Century ◽  
A Finite Group

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to RI×C for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra R[G] of a finite group G to its representation algebra R(G,R), via the natural duality of the topological vector space RI to the vector space R(I), for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups.

Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations

2002 ◽  
Vol 54 (4) ◽  
pp. 795-827 ◽  
Author(s):  
Rögnvaldur G. Möller
Keyword(s):  
Structure Theory ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Basic Concepts ◽  
Fundamental Theorems ◽  
Totally Disconnected ◽  
Permutation Actions

AbstractWillis's structure theory of totally disconnected locally compact groups is investigated in the context of permutation actions. This leads to new interpretations of the basic concepts in the theory and also to new proofs of the fundamental theorems and to several new results. The treatment of Willis's theory is self-contained and full proofs are given of all the fundamental results.

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