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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2019, Issue 754

# Kazhdan projections, random walks and ergodic theorems

Cornelia Druţu
/ Piotr W. Nowak
• Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland; and Institute of Mathematics, University of Warsaw, Poland
• Email
• Other articles by this author:
Published Online: 2017-03-18 | DOI: https://doi.org/10.1515/crelle-2017-0002

## 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.

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Revised: 2016-12-24

Published Online: 2017-03-18

Published in Print: 2019-09-01

Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: Geometric and analytic aspects of infinite groups

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ANR Blanc ANR-10-BLAN 0116

Award identifier / Grant number: Labex CEMPI ANR-11-LABX-0007-01

Funding Source: Narodowe Centrum Nauki

Award identifier / Grant number: DEC-2013/10/EST1/00352

The research of both authors was supported by the EPSRC grant “Geometric and analytic aspects of infinite groups”. The research of the first author was also partially supported by the project ANR Blanc ANR-10-BLAN 0116, acronym GGAA, and by the Labex CEMPI (ANR-11-LABX-0007-01). The research of the second author was partially supported by Narodowe Centrum Nauki grant DEC-2013/10/EST1/00352.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 754, Pages 49–86, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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