Warped cones, (non)rigidity, and piecewise properties, with a joint appendix with Dawid Kielak
Abstract
We prove that if a quasiisometry 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 quasiisometry of the respective warped cones. For a general quasiisometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasiisometric after taking Cartesian products with suitable powers of the integers. Secondly, 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 quasiisometries by showing that quasiisometric warped cones need not come from quasiisometric groups, contrary to the case of box spaces.
 Publication:

arXiv eprints
 Pub Date:
 July 2017
 DOI:
 10.48550/arXiv.1707.02960
 arXiv:
 arXiv:1707.02960
 Bibcode:
 2017arXiv170702960S
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Dynamical Systems;
 Mathematics  Group Theory;
 Mathematics  Operator Algebras
 EPrint:
 new Corollary 8.8 answering Willett's question on possible values of dynamic asymptotic dimension