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Abstracts of the lectures
Gilles **Pisier**. *Uniform convexity and related metric
geometric properties of
Banach spaces
*

Abstract
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 l*^{p}-cohomology of
finitely generated groups

In a first time,
l^{p}-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 l^{p}-cohomology of
*G*
is
trivial. In a second time the talk will focus on l^{p}-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 l^{p}
for
*p* large enough.

Abstracts of the talks
Pierre **Pansu**. *Products in L*^{p}-cohomology and
curvature
pinching

A Künneth formula is used to compute
L^{p}-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 L^{p} space or
any Banach space isomorphic to a Hilbert space. In this talk, we will show
that the universal lattice
SL_{n}(**Z**[X_{1},...,X_{k}]) has a fixed
point property for affine isometric actions on *B*. We will also
verify that the comparison map in degree two

Ψ^{2}:
H^{2}_{b}(SL_{n}(**Z**[X_{1},...,X_{k}]),*B*)→
H^{2}(SL_{n}(**Z**[X_{1},...,X_{k}]),*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**. *L*^{p}-cohomology of Lie
groups

I will give both an algebraic and a geometric characterization of
connected
Lie groups with non-trivial reduced L^{p}-cohomology in degree
one. A similar
characterization holds for algebraic groups over **Q**_{p}.
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 **Z**^{2}
is left-orderable,
though
it has the relative property (T) (with respect to
**Z**^{2}).

Groups of C^{3/2+}-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 L^{p}-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 **Z**^{2} into
the group of C^{1}-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*=SL_{3}(**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
SL_{3}(**R**) and SL_{3}(**C**), to expanders with
large girth and to Banach space of finite cotype?
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