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

Sawicki D, Kielak D (2019)
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY 118(4): 753-786.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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 Szabo, 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.
Erscheinungsjahr
2019
Zeitschriftentitel
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
Band
118
Ausgabe
4
Seite(n)
753-786
ISSN
0024-6115
eISSN
1460-244X
Page URI
https://pub.uni-bielefeld.de/record/2934992

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Sawicki D, Kielak D. Warped cones, (non-)rigidity, and piecewise properties. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY. 2019;118(4):753-786.
Sawicki, D., & Kielak, D. (2019). Warped cones, (non-)rigidity, and piecewise properties. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 118(4), 753-786. doi:10.1112/plms.12192
Sawicki, D., and Kielak, D. (2019). Warped cones, (non-)rigidity, and piecewise properties. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY 118, 753-786.
Sawicki, D., & Kielak, D., 2019. Warped cones, (non-)rigidity, and piecewise properties. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 118(4), p 753-786.
D. Sawicki and D. Kielak, “Warped cones, (non-)rigidity, and piecewise properties”, PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, vol. 118, 2019, pp. 753-786.
Sawicki, D., Kielak, D.: Warped cones, (non-)rigidity, and piecewise properties. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY. 118, 753-786 (2019).
Sawicki, Damian, and Kielak, Dawid. “Warped cones, (non-)rigidity, and piecewise properties”. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY 118.4 (2019): 753-786.