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.
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
Funding statement: 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.
Acknowledgements
The second author would like to thank the Mathematical Institute at the University of Oxford for its hospitality during a four-month stay which made this work possible. Both authors thank Mikael de la Salle, Adam Skalski, Alain Valette, Rufus Willett and Guoliang Yu for valuable comments. We are grateful to the referee for numerous suggestions that allowed to improve the presentation significantly.
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