Kazhdan projections, random walks and ergodic theorems

@article{Drutu2019KazhdanPR,
  title={Kazhdan projections, random walks and ergodic theorems},
  author={Cornelia Drutu and Piotr W. Nowak},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={2019},
  volume={2019},
  pages={49 - 86}
}
  • Cornelia Drutu, P. Nowak
  • Published 14 January 2015
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
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… 
View PDF on arXiv
31 Citations
Highly Influential Citations
5
Background Citations
9
Methods Citations
4
Results Citations
1

Figures from this paper

31 Citations

A local characterization of Kazhdan projections and applications
We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group G. We also define and…
ASYMPTOTIC EXPANDERS, DOMAINS OF EXPANSION AND STRONG ERGODICITY
In this paper we initiate a geometric study of strong ergodicity. We use these techniques to prove a structure result (strongly ergodic actions admits exhaustions by domains of expansion); we show…
Superexpanders from group actions on compact manifolds
It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the…
On strong property (T) and fixed point properties for Lie groups
We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a…
On some coarse geometric notions inspired by topology and category theory
TLDR
This thesis introduces and study coarse versions of the following classical topological notions: connectedness, monotone-light factorizations, extension theorems, and quotients by properly discontinuous group actions.
Warped cones over profinite completions
We construct metric spaces that do not have property A yet are coarsely embeddable into the Hilbert space. Our examples are so-called warped cones, which were introduced by J. Roe to serve as…
Warped cones and proper affine isometric actions of discrete groups on Banach spaces
Warped cones are metric spaces introduced by John Roe from discrete group actions on compact metric spaces to produce interesting examples in coarse geometry. We show that a certain class of warped…
Higher Kazhdan projections, $\ell_2$-Betti numbers and Baum-Connes conjectures
We introduce higher-dimensional analogs of Kazhdan projections in matrix algebras over group $C^*$-algebras and Roe algebras. These projections are constructed in the framework of cohomology with…
Averaged projections, angles between groups and strengthening of Banach property (T)
Recently, Lafforgue introduced a new strengthening of Banach property (T), which he called strong Banach property (T) and showed that this property has implications regarding fixed point properties…
Analysis on simple Lie groups and lattices
. We present a simple tool to perform analysis with groups such as SL n ( R ) and SL n ( Z ), that has been introduced by Vincent Laf-forgue in his study of non-unitary representations, in connection…
...
...

References

SHOWING 1-10 OF 139 REFERENCES
COUNTEREXAMPLES TO THE COARSE BAUM-CONNES CONJECTURE
Let X be a discrete, bounded geometry metric space. Associated to X is a C∗-algebra C∗(X) which has proved very useful in C∗-algebraic approaches to the Novikov conjecture in manifold theory. The…
The Baum-Connes conjecture with coefficients for word-hyperbolic groups (after Vincent Lafforgue)
Already in the early eighties, A. Connes emphasized that Kazhdan’s property (T), which means that the trivial representation of a locally compact group is separated from all other unitary…
Chapter 13 - Pointwise Ergodic Theorems for Actions of Groups
Quantitative ergodic theorems and their number-theoretic applications
We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral…
Kazhdan and Haagerup properties from the median viewpoint
Towards strong Banach property (T) for SL(3,ℝ)
We prove that SL(3, ℝ) has Strong Banach property (T) in Lafforgue’s sense with respect to the Banach spaces that are θ > 0 interpolation spaces (for the complex interpolation method) between an…
Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T)
Let k be any locally compact non-discrete field. We show that finite invariant measures for k-algebraic actions are obtained only via actions of compact groups. This extends both Borel's density and…
Higher index theory for certain expanders and Gromov monster groups I
Random groups, random graphs and eigenvalues of p-Laplacians
Property (T) and rigidity for actions on Banach spaces
We study property (T) and the fixed-point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that…
...
...