Metric Geometry of Locally Compact Groups

Cornulier, Yves de; Harpe, Pierre de la

doi:10.4171/166

Cited by 120 publications

(144 citation statements)

References 110 publications

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“…We observe that supramenability is invariant under coarse equivalence between metric spaces with bounded geometry as this is the case for amenability (see [, Corollary 2.2 and Theorem 3.1; , Proposition 3.D.33]).…”

Section: Properly Infinite Projections In Uniform Roe Algebrasmentioning

confidence: 94%

“…Note that by [, Theorem 32] a bounded geometry metric space is paradoxical if and only if it is non‐amenable, where we recall that a bounded geometry metric space is amenable if for any

r, ε > 0

there exists a finite subset

F \subseteq X

such that

| {x \in X ∣ d (x, F) ⩽ r} | ⩽ (1 + ε) | F | .

In particular, it follows that if

A

and

B

are coarsely equivalent in the sense of Definition , then

A

is paradoxical if and only if

B

is (see, for example, [, Corollary 2.2 and Theorem 3.1; , Proposition 3.D.33]).…”

Section: Properly Infinite Projections In Uniform Roe Algebrasmentioning

confidence: 99%

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Low‐dimensional properties of uniform Roe algebras

Willett

2018

J. London Math. Soc.

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Abstract. The goal of this paper is to study when uniform Roe algebras have certain C * -algebraic properties in terms of the underlying space: in particular, we study properties like having stable rank one or real rank zero that are thought of as low dimensional, and connect these to low dimensionality of the underlying space in the sense of the asymptotic dimension of Gromov. Some of these results (for example, on stable rank one, cancellation, strong quasidiagonality, and finite decomposition rank) give definitive characterizations, while others (on real rank zero) are only partial and leave a lot open.We also establish results about K-theory, showing that all K0-classes come from the inclusion of the canonical Cartan in low dimensional cases, but not in general; in particular, our K-theoretic results answer a question of Elliott and Sierakowski about vanishing of K0 groups for uniform Roe algebras of non-amenable groups. Along the way, we extend some results about paradoxicality, proper infiniteness of projections in uniform Roe algebras, and supramenability from groups to general metric spaces. These are ingredients needed for our K-theoretic computations, but we also use them to give new characterizations of supramenability for metric spaces. Mathematics Subject Classification (2010): 20F69, 46L05, 46L80, 55M10

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Section: Properly Infinite Projections In Uniform Roe Algebrasmentioning

confidence: 94%

“…Note that by [, Theorem 32] a bounded geometry metric space is paradoxical if and only if it is non‐amenable, where we recall that a bounded geometry metric space is amenable if for any

r, ε > 0

there exists a finite subset

F \subseteq X

such that

| {x \in X ∣ d (x, F) ⩽ r} | ⩽ (1 + ε) | F | .

In particular, it follows that if

A

and

B

are coarsely equivalent in the sense of Definition , then

A

is paradoxical if and only if

B

is (see, for example, [, Corollary 2.2 and Theorem 3.1; , Proposition 3.D.33]).…”

Section: Properly Infinite Projections In Uniform Roe Algebrasmentioning

confidence: 99%

Low‐dimensional properties of uniform Roe algebras

Willett

2018

J. London Math. Soc.

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“…The group

\overset{H}{true}

is compactly generated since it contains

G

as a cocompact subgroup. In view of [, Proposition 8.A.10],

ker (β)

is the normal closure of a compact set. Applying Theorem ,

ker (β)

is discrete and contained in the

F C

‐center of

\overset{H}{true}

, hence

ker (β)

is indeed finitely generated by some finite subset

X

.…”

Section: Factorization Of Normal Compressions In Lcsc Groupsmentioning

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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“…For example, every connected and simply connected Lie group is compactly presented. Many interesting examples of compactly presented groups and their properties are discussed in [CdlH16,Chapter 8]. It is interesting that compact presentability of topological groups can be used to prove finite presentability of discrete groups; see [CdlH16, Section 1.E].…”

Section: −1mentioning

confidence: 99%

“…Sections 4.D and 4.E of [CdlH16] give examples of such topics (locally elliptic groups and capped subgroups).…”

Section: −1mentioning

confidence: 99%