Cantor systems and quasi-isometry of groups

Medynets, Kostya; Sauer, Roman; Thom, Andreas

doi:10.1112/blms.12059

Cited by 10 publications

(7 citation statements)

References 20 publications

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“…Let us now observe that nuclear Roe algebras have distinguished Cartan subalgebras (see Theorem 6.2). This follows from [57] and [32, § 2.5] (see also [36]). We refer to [57,32] and the references therein for background material concerning notions from geometric group theory such as quasi-isometry or bilipschitz equivalence.…”

Section: Cartan Subalgebras In Roe Algebrasmentioning

confidence: 75%

Cartan subalgebras in C*-algebras. Existence and uniqueness

Renault

2019

Trans. Amer. Math. Soc.

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We initiate the study of Cartan subalgebras in C*-algebras, with a particular focus on existence and uniqueness questions. For homogeneous C*-algebras, these questions can be analysed systematically using the theory of fibre bundles. For group C*-algebras, while we are able to find Cartan subalgebras in C*-algebras of many connected Lie groups, there are classes of (discrete) groups, for instance non-abelian free groups, whose reduced group C*-algebras do not have any Cartan subalgebras. Moreover, we show that uniqueness of Cartan subalgebras usually fails for classifiable C*-algebras. However, distinguished Cartan subalgebras exist in some cases, for instance in nuclear Roe algebras.

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Section: Cartan Subalgebras In Roe Algebrasmentioning

confidence: 75%

Cartan subalgebras in C*-algebras. Existence and uniqueness

Renault

2019

Trans. Amer. Math. Soc.

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“…Z 2 * Z 3 by a topologically free action on a Cantor set. By Theorem 3.2 in [27] (in fact, by the identically-proved version of this result with "topological freeness" instead of "freeness"), and since these free products are clearly non-amenable, it follows that they are quasi-isometric. Now the result follows from Lemma 7.8.…”

Section: Tensor Products Of L P -Cuntz Algebrasmentioning

confidence: 91%

“…H implies G ∼ bi−L H, while the converse is known to be false in general. However, the converse does hold if either G or H is non-amenable [27].…”

Section: Tensor Products Of L P -Cuntz Algebrasmentioning

confidence: 99%

Rigidity results for $L^p$-operator algebras and applications

Choi¹,

Gardella²,

Thiel³

2019

Preprint

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For p ∈ [1, ∞), we show that every unital L p -operator algebra contains a unique maximal C * -subalgebra, which we call the C * -core. When p = 2, the C * -core of an L p -operator algebra is abelian. Using this, we canonically associate to every unital L p -operator algebra A an étale groupoid G A , which in many cases of interest is a complete invariant for A. By calculating this groupoid for large classes of examples, we obtain a number of rigidity results that display a stark contrast with the case p = 2; the most striking one being that of crossed products by essentially free actions.These rigidity results give answers to questions concerning the existence of homomorphisms or isomorphisms between different algebras. Among others, we show that for the L p -analog O p 2 of the Cuntz algebra, there is no isometric isomorphism between O p 2 and O p 2 ⊗p O p 2 , when p = 2. In particular, we deduce that there is no L p -version of Kirchberg's absorption theorem, and that there is no K-theoretic classification of purely infinite simple amenable L p -operator algebras for p = 2.

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“…The idea of developing dynamic characterizations of coarse equivalence goes back to Gromov's notion of topological couplings and has been developed further in [43,41]. Recently, independently from the author, a dynamic characterization of bilipschitz equivalence for finitely generated groups was obtained in [31], which is a special case of our result.…”

Section: Introductionmentioning

confidence: 75%

Dynamic characterizations of quasi-isometry and applications to cohomology

2018

Algebr. Geom. Topol.

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We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we give conceptual explanations for previous results of Shalom and Sauer on coarse invariance of homological and cohomological dimensions and Shalom's property H FD . As another application, we show that group homology and cohomology in a class of coefficients, including all induced and co-induced modules, are coarse invariants. We deduce that being of type FP n (over arbitrary rings) is a coarse invariant, and that being a (Poincaré) duality group over a ring is a coarse invariant among all groups which have finite cohomological dimension over that ring. Our results also imply that every self coarse embedding of a Poincaré duality group over an arbitrary ring must be a coarse equivalence. It is then evident that coarse equivalences induce isomorphisms as they are precisely those coarse maps which are invertible modulo ∼. A similar remark applies to cohomology. Thus, not only these (co)homology groups, but, by functoriality, the actions of the groups of coarse equivalences (modulo ∼) on these (co)homology groups are coarse invariants as well. We obtain analogous results for coarse embeddings in the topological setting, i.e., for topological res-invariant modules and reduced (co)homology. It turns out that coarse embeddings always induce isomorphisms in (co)homology and reduced (co)homology.The aforementioned results on coarse invariance of type FP n and being a (Poincaré) duality group are immediate consequences, as is our rigidity result for coarse embeddings into Poincaré duality groups. We also deduce that vanishing of ℓ 2 -Betti numbers is a coarse invariant, as observed in [35,34,32], and generalized by Sauer and Schrödl to all unimodular locally compact second countable groups [42]. This is a good point to formulate an interesting and natural question, which we elaborate on in § 4.4:Question (Question 4.44). Are homological and cohomological dimension over a commutative ring R with unit always coarse invariants among all countable discrete groups with no R-torsion?We refer to § 4 for more details. § 3 and § 4 are independent from each other. Thus readers interested in this last set of results on coarse invariance of group (co)homology may go directly from § 2 to § 4.As far as our methods are concerned, we use groupoid techniques as in [43,41,34], but instead of working with abstract dynamical systems, we base our work on concrete dynamic characterizations of coarse equivalence. The difference between our work and [19] is that we do not work with descriptions of group (co)homology in terms of Eilenberg-MacLane spaces, as these descriptions require finiteness conditions (like F n or F ∞ ) on our groups and have to be modified whenever we change coefficients. Instead, since coarse embeddings automatically lead to "controlled" orbit equivalences satisfying the finiteness condition mentioned above, we can work directly with...

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Cantor systems and quasi-isometry of groups

Cited by 10 publications

References 20 publications

Cartan subalgebras in C*-algebras. Existence and uniqueness

Cartan subalgebras in C*-algebras. Existence and uniqueness

Rigidity results for $L^p$-operator algebras and applications

Dynamic characterizations of quasi-isometry and applications to cohomology

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