Preserving coarse properties

Dydak, Jerzy; Virk, Žiga

doi:10.1007/s13163-015-0182-x

Cited by 10 publications

(31 citation statements)

References 12 publications

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“…Note that by Proposition 3.8 in this paper and Corollary 9.4 in [7], X has Property A if X G does, for any countable group G. The following corollary was already proved in [6].…”

Section: The Maximal Roe Algebramentioning

confidence: 55%

Coarse quotients by group actions and the maximal Roe algebra

Higginbotham

Weighill

2019

J. Topol. Anal.

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For a discrete metric space (or more generally a large scale space) X and an action of a group G on X by coarse equivalences, we define a type of coarse quotient space X G , which agrees up to coarse equivalence with the orbit space X G when G is finite. We then restrict our attention to what we call coarsely discontinuous actions and show that for such actions the group G can be recovered as an appropriately defined automorphism group Aut(X X G ) when X satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group G on a discrete bounded geometry metric space X there is a relation between the maximal Roe algebras of X and X G , namely that there is a * -isomorphism C * max (X G ) K ≅ C * max (X) K⋊G, where K is the ideal of compact operators. If X has Property A and G is amenable, then we show that X G has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

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“…Note that by Proposition 3.8 in this paper and Corollary 9.4 in [7], X has Property A if X G does, for any countable group G. The following corollary was already proved in [6].…”

Section: The Maximal Roe Algebramentioning

confidence: 55%

Coarse quotients by group actions and the maximal Roe algebra

Higginbotham

Weighill

2019

J. Topol. Anal.

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“…Several authors have studied the so-called permanence properties -that is, the extent to which properties are preserved by unions, products, etc -of coarse invariants such as finite asymptotic dimension [3], finite decomposition complexity [19,15], or property A [7]. The permanence properties of asymptotic property C have proved to be slightly more elusive than others, with incremental special cases being proved for unions [5] and certain types of group extensions and products [1,2].…”

Section: Direct Products Preserve Asymptotic Property Cmentioning

confidence: 99%

On the stability of asymptotic property C for products and some group extensions

Bell

Nagórko²

2018

Algebr. Geom. Topol.

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We show that Dranishnikov's asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if G and H are groups with asymptotic property C, then both G×H and G * H have asymptotic property C. We also prove that a group G has asymptotic property C if 1 → K → G → H → 1 is exact, if asdim K < ∞, and if H has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk [15], of Bell and Moran [5], and an open problem in topology in [23].2010 Mathematics Subject Classification. 54F45 (primary), 20F69 (secondary).

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“…Proof. As in Example 4.2 of [14] we may assume that G acts on X by isometries. According to Proposition 3.1 of [20] G induces an action on νX and the map induced by X → G \ X is the quotient map νX → G \ νX, which is open.…”

Section: 2mentioning

confidence: 99%

“…• coarsely surjective as it is surjective, • coarse by the definition of the Hausdorff metric on G \ X, • coarsely |G|-to-1 by [14], • coarsely open by Proposition 4.17.…”

Section: Dimension Preserving Theorem For Coarsely Open Coarsely N-tomentioning

confidence: 99%

“…They may be used to classify the asymptotic dimension of a space [24]. Furthermore, if G is a finite group acting on a metric space Z by coarse maps then the quotient Z → G\Z is coarse and coarsely n-to-1, see Example 4.2 in [14]. Miyata and Virk introduced these maps in [24] and established analogues of classical results for n-to-1 maps in topology.…”

Section: Introductionmentioning

confidence: 99%

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Higson compactification and dimension raising

Austin

Virk

2017

Topology and its Applications

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Let X and Y be proper metric spaces. We show that a coarsely n-to-1 map f : X → Y induces an n-to-1 map of Higson coronas. This viewpoint turns out to be successful in showing that the classical dimension raising theorems hold in large scale; that is, if f : X → Y is a coarsely n-to-1 map between proper metric spaces X and Y then asdim(Y ) ≤ asdim(X) + n − 1. Furthermore we introduce coarsely open coarsely n-to-1 maps, which include the natural quotient maps via a finite group action, and prove that they preserve the asymptotic dimension.

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Preserving coarse properties

Cited by 10 publications

References 12 publications

Coarse quotients by group actions and the maximal Roe algebra

Coarse quotients by group actions and the maximal Roe algebra

On the stability of asymptotic property C for products and some group extensions

Higson compactification and dimension raising

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