Homomorphisms into totally disconnected, locally compact groups with dense image

Reid, Colin D.; Wesolek, Phillip

doi:10.1515/forum-2018-0017

Cited by 11 publications

(10 citation statements)

References 12 publications

(12 reference statements)

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“…groups and the theory of Hecke pairs, that is, groups with a specified commensurated subgroup. (For a general discussion of this connection, see [47].) An example of this connection is given by Proposition 3.6 below.…”

Section: Connections Between Topological and Abstract Simplicitymentioning

confidence: 99%

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Caprace

Reid

Willis

2017

Forum of Mathematics, Sigma

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We use the structure lattice, introduced in Part I, to undertake a systematic study of the class S consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are nondiscrete. Given G ∈ 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 S abstractly simple? Can a group in S be amenable? Can a group in S be such that the contraction groups of all of its elements are trivial? 2010 Mathematics Subject Classification: 22D05 (primary); 20E15, 20E32 (secondary)

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Section: Connections Between Topological and Abstract Simplicitymentioning

confidence: 99%

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Caprace

Reid

Willis

2017

Forum of Mathematics, Sigma

Self Cite

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“…In particular, this canonical quotient map H ⋌ L → H ⋌ Schl L has a compact kernel. (See for instance [RW,§4] for these facts. )…”

Section: Haagerup and Pw Propertiesmentioning

confidence: 99%

“…In particular, this canonical quotient map H L → H Schl L has a compact kernel. (See, for instance, [32,Section 4] for these facts.) (c) The Schlichting completion is sometimes called the relative profinite completion.…”

Section: Commensurated Subgroups and Relative Profinite Completionmentioning

confidence: 99%

Locally Compact Wreath Products

Cornulier¹

2018

J. Aust. Math. Soc.

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Wreath products of non-discrete locally compact groups are usually not locally compact groups, nor even topological groups. We introduce a natural extension of the wreath product construction to the setting of locally compact groups.As an application, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without nontrivial compact normal subgroups.

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“…For a discrete group H, let φ : H → G be a group homomorphism with dense image in G. Such a homomorphism is referred to as a totally disconnected locally compact completion of H (a general framework for totally disconnected locally compact completions can be found in [12]). Note that φ is not required to be injective.…”

Section: Subcase 1amentioning

confidence: 99%

A property of the lamplighter group

Castellano

Cook

Kropholler

2020

Topological Algebra and Its Applications

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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.A subgroup H of a group G is said to be inert if H and g −1 Hg are commensurate for all g ∈ G, meaning that H ∩ H g always has finite index in both H and H g . The terminology was introduced by Kegel and has been explored in many contexts (see, for example, the recent survey [8]). In abstract group theory, Robinson's investigation [13] focusses on soluble groups. Here we study a particular special family of soluble groups: the lamplighter groups and our interest is in the connection with the theory of totally disconnected locally compact groups. In that context, inert subgroups are particularly important in the light of van Dantzig's theorem that every totally disconnected locally compact group has a compact open subgroup and of course all such subgroups are commensurate with one another and therefore inert. It should be noted that in recent literature it is common to use the term commensurated in place of inert, see for example [5,6,7,10]. In §2 we remark a dynamical aspect of the property investigated here and relate it to the concept of algebraic entropy.The relation of commensurability is an equivalence relation amongst the subgroups of a group. By a class we shall here mean an equivalence class of subgroups under this relation. For a prime p, the corresponding lamplighter group is the standard restricted wreath product F p wr Z, i.e., the standard restricted wreath product of a group of order p by an infinite cyclic group. These are the simplest of soluble groups that fall outside the classes considered by Robinson [13]. Our main observation is as follows.Theorem. The inert subgroups of the lamplighter group F p wr Z fall into exactly five classes.

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Homomorphisms into totally disconnected, locally compact groups with dense image

Cited by 11 publications

References 12 publications

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Locally Normal Subgroups of Totally Disconnected Groups. Part Ii: Compactly Generated Simple Groups

Locally Compact Wreath Products

A property of the lamplighter group

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