Decomposing locally compact groups into simple pieces

Caprace, Pierre–Emmanuel; Monod, Nicolas

doi:10.1017/s0305004110000368

Cited by 67 publications

(97 citation statements)

References 42 publications

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“…The group

G

is then a t.d.l.c.s.c. group, and via [, Proposition 4.3],

G

is regionally SIN. Lemma now ensures that

ξ (G) ⩽ 2

□

…”

Section: Elementary Groups and The Decomposition Rankmentioning

confidence: 99%

Dense normal subgroups and chief factors in locally compact groups

Reid

Wesolek

2017

Proc. London Math. Soc.

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In The essentially chief series of a compactly generated locally compact group, an analogue of chief series for finite groups is discovered for compactly generated locally compact groups. In the present article, we show that chief factors necessarily exist in all locally compact groups with sufficiently rich topological structure. We also show that chief factors have one of seven types, and for all but one of these types, there is a decomposition into discrete groups, compact groups, and topologically simple groups. Our results for chief factors require exploring the theory developed in Chief factors in Polish groups in the setting of locally compact groups. In this context, we obtain tighter restrictions on the factorization of normal compressions and the structure of quasi‐products. Consequently, both (non‐)amenability and elementary decomposition rank are preserved by normal compressions.

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“…The group

G

is then a t.d.l.c.s.c. group, and via [, Proposition 4.3],

G

is regionally SIN. Lemma now ensures that

ξ (G) ⩽ 2

□

…”

Section: Elementary Groups and The Decomposition Rankmentioning

confidence: 99%

Dense normal subgroups and chief factors in locally compact groups

Reid

Wesolek

2017

Proc. London Math. Soc.

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“…More subtly still, a residually discrete t.d.l.c.s.c. group may be written as a countable increasing union of small invariant neighbourhood (SIN) groups by results of Caprace and Monod [7]. t.d.l.c.s.c.…”

Section: Introductionmentioning

confidence: 99%

Elementary totally disconnected locally compact groups

Wesolek

2015

Proceedings of the London Mathematical Society

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We identify the class of elementary groups: the smallest class of totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contains the profinite groups and the discrete groups, is closed under group extensions of profinite groups and discrete groups, and is closed under countable increasing unions. We show this class enjoys robust permanence properties. In particular, it is closed under group extension, taking closed subgroups, taking Hausdorff quotients, and inverse limits. A characterization of elementary groups in terms of well-founded descriptive-set-theoretic trees is then presented. We conclude with three applications. We first prove structure results for general t.d.l.c.s.c. groups. In particular, we show a compactly generated t.d.l.c.s.c. group decomposes into elementary groups and topologically characteristically simple groups via group extension. We then prove two local-to-global structure theorems: Locally solvable t.

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“…Let G be a compactly generated totally disconnected locally compact group without non-trivial compact or discrete normal subgroup. Then: (i) every non-trivial closed normal subgroup contains a minimal one; (ii) the set M of non-trivial minimal closed normal subgroups need not be finite, but the subset M na of its non-abelian elements is finite; (iii) each abelian M ∈ M is topologically locally finite, hence contained in RadAssertions (i) and (iv) are stated and correctly proved in [1]. The erroneous assertion in loc.…”

mentioning

confidence: 95%

“…(Received 9 February 2015; revised 7 August 2017) Phillip Wesolek (personal communication) pointed out an error in Proposition 2·6 from our paper [1]; we thank him for his perspicacious remark, as well as for helpful comments on an earlier version of this correction. That error is corrected below; as we shall see, it only affects the statements of Theorem B and Proposition 2·6.…”

mentioning

confidence: 96%

Corrigendum to “Decomposing locally compact groups into simple pieces” [Math. Proc. Camb. Phil. Soc. 150 (1) (2011) 97-128]

Caprace¹,

Monod²

2017

Math. Proc. Camb. Phil. Soc.

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(personal communication) pointed out an error in Proposition 2·6 from our paper [1]; we thank him for his perspicacious remark, as well as for helpful comments on an earlier version of this correction. That error is corrected below; as we shall see, it only affects the statements of Theorem B and Proposition 2·6. An independent inaccurracy in [1, Theorem E(ii)], pointed out to us by Alain Valette (personal communication), is also rectified below. All other results of the original paper hold without change.We retain all the notation and conventions from [1]; moreover any numbered statement quoted here refers to the corresponding statement from loc. cit. The invalid claim is that the intersection N is trivial, in the last sentence of the proof of Proposition 2·6. That proposition must be replaced by the following: PROPOSITION 2·6 (corrected). Let G be a compactly generated totally disconnected locally compact group without non-trivial compact or discrete normal subgroup. Then: (i) every non-trivial closed normal subgroup contains a minimal one; (ii) the set M of non-trivial minimal closed normal subgroups need not be finite, but the subset M na of its non-abelian elements is finite; (iii) each abelian M ∈ M is topologically locally finite, hence contained in RadAssertions (i) and (iv) are stated and correctly proved in [1]. The erroneous assertion in loc. cit. is that the whole set M is finite. Indeed, a counter-example is provided by the group (Q p ) n

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Decomposing locally compact groups into simple pieces

Cited by 67 publications

References 42 publications

Dense normal subgroups and chief factors in locally compact groups

Dense normal subgroups and chief factors in locally compact groups

Elementary totally disconnected locally compact groups

Corrigendum to “Decomposing locally compact groups into simple pieces” [Math. Proc. Camb. Phil. Soc. 150 (1) (2011) 97-128]

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