Amenable groupoids

Anantharaman, Claire; Renault, Jean

doi:10.1090/conm/282/04677

Cited by 148 publications

(336 citation statements)

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“…In this subsection, we recall the construction of weakly orbit equivalent actions from an ME coupling given in Section 3 in [13]. We refer the reader to [1] and Chapter XIII, Section 3 in [41] for the terminology of a discrete measured groupoid and its amenability. We fix the notation as follows: Given a discrete measured groupoid Ᏻ on a standard finite measure space .X; / (i.e., a standard Borel space with a finite positive measure) and a Borel subset A X with positive measure, we denote by .Ᏻ/ A D f 2 Ᏻ j r. /; s. / 2 Ag the groupoid restricted to A, where r; sW Ᏻ !…”

Section: 2mentioning

confidence: 99%

Measure equivalence rigidity of the mapping class group

Kida

2010

Ann. Math.

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We show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence: If the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernels. Moreover, we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group. We obtain similar results for finite direct products of mapping class groups.

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Section: 2mentioning

confidence: 99%

Measure equivalence rigidity of the mapping class group

Kida

2010

Ann. Math.

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“…Assume that X is a topologically amenable G-space (in the sense of [1]) and that A is any X G-algebra. Then G satisfies BC for A.…”

Section: Lemma 38 Assume a Is An X G-algebra For Some Proper G-spacmentioning

confidence: 99%

“…Tu was also able to prove a groupoid version of the above mentioned result of Higson and Kasparov (see [21]). In particular, all topologically amenable groupoids (in the sense of [1]) satisfy BC for arbitrary coefficients.…”

mentioning

confidence: 99%

Shapiro's lemma for topological K-theory of groups

Chabert¹,

Echterhoff²,

Oyono‐Oyono³

2003

Comment. Math. Helv.

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Abstract. Let X G be the crossed product groupoid of a locally compact group G acting on a locally compact space X. For any X G-algebra A we show that a natural forgetful map from the topological K-theory K top * (X G; A) of the groupoid X G with coefficients in A to the topological K-theory K top * (G; A) of G with coefficients in A is an isomorphism. We then discuss several interesting consequences of this result for the Baum-Connes conjecture.

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“…Proba.G/ such that lim n!1 sup x2X kgm n x m n gx k 1 D 0 for all g 2 G (cf. [14], [10], [2]). With this definition, a group is amenable if and only if the action on an one-point space is Zimmer amenable, while such an action is always Greenleaf amenable.…”

Section: Introductionmentioning

confidence: 99%

“…The group 2 can be viewed as an amalgamated free product of two copies of F 2 along the subgroup generated by the commutator, i.e., 2 Theorem. The surface group g belongs to A for all g 2.…”

Section: Introductionmentioning

confidence: 99%

Amenable actions of amalgamated free products

Moon¹

2010

Groups Geom. Dyn.

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Abstract. We prove that the amalgamated free product of two free groups of rank two over a common cyclic subgroup admits an amenable, faithful, transitive action on an infinite countable set. We also show that any finite index subgroup admits such an action, which applies for example to surface groups and fundamental groups of surface bundles over S 1 .Mathematics Subject Classification (2010). 43A07, 20E06, 57M07.

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Amenable groupoids

Cited by 148 publications

References 0 publications

Measure equivalence rigidity of the mapping class group

Measure equivalence rigidity of the mapping class group

Shapiro's lemma for topological K-theory of groups

Amenable actions of amalgamated free products

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