scholarly journals Compact and compactly generated subgroups of locally compact groups

1990 ◽  
Vol 108 (4) ◽  
pp. 1085-1085
Author(s):  
R. W. Bagley ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Compactly Generated

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.

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?

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 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 A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Colin D. Reid
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Derived Length ◽  
Chief Factors ◽  
Compactly Generated

AbstractWe classify the locally compact second-countable (l.c.s.c.) groups 𝐴 that are abelian and topologically characteristically simple. All such groups 𝐴 occur as the monolith of some soluble l.c.s.c. group 𝐺 of derived length at most 3; with known exceptions (specifically, when 𝐴 is \mathbb{Q}^{n} or its dual for some n\in\mathbb{N}), we can take 𝐺 to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups.

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