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Abstracts of the lectures

Gilles Pisier. Uniform convexity and related metric geometric properties of Banach spaces


Alain Valette. Property (T) and harmonic maps: Gelfand pairs and Sp(n,1)

A locally compact, σ-compact group G has property (T) if every isometric action of G on a Hilbert space, has a fixed point. In the first lecture, we will explain a result by Shalom that, for Lie groups G admitting a Gelfand pair (G,K), property (T) is equivalent to the fact that every G-equivariant harmonic map from G/K to a Hilbert space, is constant. In the second lecture, we will explain how it was used by Gromov to give the first geometric proof of property (T) for the rank one group Sp(n,1), by looking at growth of harmonic maps on G/K.

Marc Bourdon. On the lp-cohomology of finitely generated groups

In a first time, lp-homology and cohomology of simplicial complexes and of finitely generated groups will be defined. Their invariance by quasi-isometry will be presented. We will prove the following result of Gromov : if the center of a finitely generated group G is infinite, then the reduced lp-cohomology of G is trivial. In a second time the talk will focus on lp-cohomology in degree one. Harmonic representatives for non-amenable groups will be discussed. Using a construction of G. Elek, we will prove a result of G. Yu: every word hyperbolic group acts properly by isometries on lp for p large enough.

Abstracts of the talks

Pierre Pansu. Products in Lp-cohomology and curvature pinching

A Künneth formula is used to compute Lp-cohomology of complex hyperbolic plane, together with its multiplicative structure. This implies that no strictly negatively 1/4th-pinched manifold can be quasiisometric to it.

Martin Kassabov. Subspace arrangements and property T

I will mainly talk about (my viewpoint at) a method for proving Property T started by Dymara and Januszkiewicz. Their original motivation came from groups actions on finite dimensional buildings, but the refined idea does not use anything more than angles between subspaces in an finite dimensional Euclidean space.
Parts of the talk are based on a work of M. Ershov and A. Jaikin.

Masato Mimura. A fixed point property and the second bounded cohomology of universal lattices on Banach spaces

Let n≥4 be an integer, and B be any Lp space or any Banach space isomorphic to a Hilbert space. In this talk, we will show that the universal lattice SLn(Z[X1,...,Xk]) has a fixed point property for affine isometric actions on B. We will also verify that the comparison map in degree two
Ψ2: H2b(SLn(Z[X1,...,Xk]),B)→ H2(SLn(Z[X1,...,Xk]),B)

from bounded to ordinary cohomology is injective. For our proof, we establish a certain implication from the relative Kazhdan-type property to the relative fixed point property on uniformly convex Banach spaces.

Cornelia Drutu. Asymptotic median structure of mapping class groups, applications to homomorphisms

Median spaces can be seen as non-discrete versions of CAT(0) cubical complexes. It turns out that every asymptotic cone of a mapping class group has a natural equivariant structure of median space. Moreover, both Kazhdan and Haagerup properties can be characterized in terms of actions of groups on median spaces. This allows to discuss homomorphisms of groups with property (T) into mapping class groups of surfaces. The talk is on joint work with I. Chatterji and F. Haglund, and with J. Behrstock and M. Sapir.

Romain Tessera. Lp-cohomology of Lie groups

I will give both an algebraic and a geometric characterization of connected Lie groups with non-trivial reduced Lp-cohomology in degree one. A similar characterization holds for algebraic groups over Qp. This is a joint work with Cornulier which builds on a previous work of Pansu.

Andrés Navas On the affine isometric actions of groups of 1-dimensional diffeomorphisms of low regularity

It is still an open question whether a finitely generated Kazhdan group can act faithfully by homeomorphisms of the real line (equivalently, it is left-orderable). This question is very subtle, as is shown by the fact that the semidirect product of SL(2,Z) by Z2 is left-orderable, though it has the relative property (T) (with respect to Z2).
Groups of C3/2+-diffeomorphisms of the circle or the interval carry a natural affine action on a Hilbert space. For lower regularity, this can be modified to an actions Lp-spaces (for large p). However, as I will explain, this does not allow dealing with the problem for orderability of Kazhdan groups.
Despite the above, I will show (by rather different methods) that there is no embedding of any finite index subgroup of the semidirect product of SL(2,Z) by Z2 into the group of C1-diffeomorphisms of the line.

Vincent Lafforgue. Strengthened Property (T) and applications.

We first give an idea of the proof of strengthened property (T) for G=SL3(F) where F is a non-archimedean local field. It says roughly that the trivial representation is isolated among representations of G with small exponential growth in Banach spaces of type >1. From it we deduce that expanders built from G do not embed coarsely in Banach spaces of type >1, and that any affine isometric action of G in such a Banach space has a fixed point. At the end we will discuss open questions : how to extend these results to SL3(R) and SL3(C), to expanders with large girth and to Banach space of finite cotype?

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