Coarse flow spaces for relatively hyperbolic groups

Bartels, Arthur

doi:10.1112/s0010437x16008216

Cited by 27 publications

(52 citation statements)

References 41 publications

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“…Consequently, we have the following dynamical result (Theorem ) about equivariant asymptotic dimension (also known as amenability dimension), which builds on and strengthens the estimates by Szabó, Wu, and Zacharias and by Bartels . This addresses two problems of Willett, see Section 8.…”

Section: Introductionsupporting

confidence: 61%

“…The following more general result is a special case of [, Corollary 1.10]. Theorem Let

Γ ↷ Y

be a free action of a virtually nilpotent group

normalΓ

on a metrisable space

Y

of finite covering dimension.…”

Section: Applications To Dynamics and C*‐dynamicsmentioning

confidence: 99%

“…Theorem 8.5 (Bartels [3]). Let Γ Y be a free action of a virtually nilpotent group Γ on a metrisable space Y of finite covering dimension.…”

Section: Applications To Dynamics and C * -Dynamicsmentioning

confidence: 99%

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Warped cones, (non‐)rigidity, and piecewise properties

Sawicki

2018

Proc. London Math. Soc.

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We prove that if a quasi‐isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi‐isometry of the respective warped cones. For a general quasi‐isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi‐isometric after taking Cartesian products with suitable powers of the integers. Second, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone, improve bounds by Szabó, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups, and give a partial answer to a question of Willett about dynamic asymptotic dimension. In the Appendix, we justify optimality of the aforementioned result on general quasi‐isometries by showing that quasi‐isometric warped cones need not come from quasi‐isometric groups, contrary to the case of box spaces.

show abstract

Section: Introductionsupporting

confidence: 61%

“…The following more general result is a special case of [, Corollary 1.10]. Theorem Let

Γ ↷ Y

be a free action of a virtually nilpotent group

normalΓ

on a metrisable space

Y

of finite covering dimension.…”

Section: Applications To Dynamics and C*‐dynamicsmentioning

confidence: 99%

See 1 more Smart Citation

Warped cones, (non‐)rigidity, and piecewise properties

Sawicki

2018

Proc. London Math. Soc.

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“…In particular, this shows that all free ergodic measurepreserving actions of countably infinite amenable groups on atomless standard probability spaces give rise to the unique hyperfinite II 1 factor. 1 The attempt to similarly determine the structure of the C * -crossed products arising from actions of countably infinite amenable groups on compact metrizable spaces has, like the general Elliott classification program for simple separable nuclear C * -algebras, been forced to contend with obstructions of a topological nature that are conditioned by the phenomenon of higher-dimensionality. Although the precise technical connections are still not so well understood, these obstructions at the C * -algebra level are closely related to Gromov's notion of mean dimension [18], which is an entropy-like invariant in topological dynamics that provides a measure of asymptotic dimension growth.…”

Section: Introductionmentioning

confidence: 99%

“…3 The main difference between almost finiteness and the Ornstein-Weiss decomposition is that the smallness of the remainder in the former is expressed using dynamical subequivalence instead of a probability measure. While towers 1 The first proof of this fact was given by Connes as an application of his celebrated result [7] that injectivity implies hyperfiniteness, whose full force is still needed to prove hyperfiniteness of the group von Neumann algebra itself. 2 Whether or not this subequivalence is itself implemented in an approximately central way is roughly what separates Z -stability from its specialization to the nuclear setting.…”

Section: Introductionmentioning

confidence: 99%

Almost Finiteness and the Small Boundary Property

Kerr

Szabó

2019

Commun. Math. Phys.

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Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and that under this hypothesis the properties of almost finiteness, comparison, and m-comparison for some nonnegative integer m are all equivalent. The proof combines an Ornstein-Weiss tiling argument with the use of zero-dimensional extensions which are measure-isomorphic over singleton fibres. These kinds of extensions are also employed to show that if every free action of a given group on a zero-dimensional space is almost finite then so are all free actions of the group on spaces with finite covering dimension. Combined with recent results of Downarowicz-Zhang and Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this implies that crossed product C * -algebras arising from free minimal actions of groups with local subexponential growth on finite-dimensional spaces are classifiable in the sense of Elliott's program. We show furthermore that, for free actions of countably infinite amenable groups, the small boundary property implies that the crossed product has uniform property Γ, which confirms the Toms-Winter conjecture for such crossed products in the minimal case.

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The Farrell–Jones Conjecture for mapping class groups

Bartels

Bestvina

2019

Invent. math.

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We prove the Farrell-Jones Conjecture for mapping class groups. The proof uses the Masur-Minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of Teichmüller space. The proof is presented in an axiomatic setup, extending the projection axioms in [14]. More specifically, we prove that the action of Mod(Σ) on the Thurston compactification of Teichmüller space is finitely F -amenable for the family F consisting of virtual point stabilizers.

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Coarse flow spaces for relatively hyperbolic groups

Cited by 27 publications

References 41 publications

Warped cones, (non‐)rigidity, and piecewise properties

Warped cones, (non‐)rigidity, and piecewise properties

Almost Finiteness and the Small Boundary Property

The Farrell–Jones Conjecture for mapping class groups

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