Kazhdan projections, random walks and ergodic theorems

Author: 

Drutu Badea, C
Nowak, P

Publication Date: 

18 March 2017

Journal: 

Journal für die reine und angewandte Mathematik

Last Updated: 

2020-11-08T10:10:05.197+00:00

Issue: 

754

Volume: 

2019

DOI: 

10.1515/crelle-2017-0002

page: 

49-86

abstract: 

In this paper we investigate generalizations of Kazhdan’s property (T) to the setting of uniformly convex Banach spaces. We explain the interplay between the existence of spectral gaps and that of Kazhdan projections. Our methods employ Markov operators associated to a random walk on the group, for which we provide new norm estimates and convergence results. This construction exhibits useful properties and flexibility, and allows to view Kazhdan projections in Banach spaces as natural objects associated to random walks on groups. We give a number of applications of these results. In particular, we address several open questions. We give a direct comparison of properties (TE) and FE with Lafforgue’s reinforced Banach property (T); we obtain shrinking target theorems for orbits of Kazhdan groups; finally, answering a question of Willett and Yu we construct non-compact ghost projections for warped cones. In this last case we conjecture that such warped cones provide counterexamples to the coarse Baum–Connes conjecture.

Symplectic id: 

685467

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article