Titres et résumés :
Brian Bowditch
Titre : Coarse median spaces (I and II)
Résumé :
We describe the notion of a ``coarse median'' on a geodesic metric space.
This satisfies the axioms of a median algebra up to bounded distance. The existence of a coarse median is invariant under quasi-isometry and might be thought of as a kind of coarse non-positive curvature condition.
One can define a ``coarse median group'' as a finitely generated group whose Cayley graph admits a coarse median. This property is closed under direct products and relative hyperbolicity. Examples are hyperbolic groups and right-angled Artin groups. Using the centroid construction of Behrstock and Minsky, one sees that the mapping class group of a compact surface is coarse median.
I aim to describe the general background to this topic, and give some examples of applications. Most of these pass via the asymptotic cone. Under a certain finite rank condition, one sees for example that the asymptotic cone embeds in a finite product of trees and is bilipschitz equivalent to a finite dimensional CAT(0) space. This has various consequences for the large scale structure of the group.
Victor Chepoi
Titre : Bucolic graphs and their relatives
Résumé :
Median graphs (1-skeletons
of CAT(0) cube complexes) satisfy several fundamental properties, some
of which characterize them: (A) median graphs are exactly the retracts
of hypercubes, (B) any non-expansive mapping on a median graph fixes a
cube, (C) finite median graphs can be obtained from cubes via gated
amalgams, (D) median graphs are the underlying graphs of discrete
median algebras, (E) cube complexes of median graphs are contractible,
(F) cube complexes of median graphs are exactly the simply
connected cube complexes satisfying the local cube condition.
To what extent, similar properties hold for more general
classes of graphs? Currently, all such known results concern subclasses
of weakly modular graphs, i.e., graphs satisfying so-called triangle
and quadrangle conditions. Results analogous to (A)-(D) were obtained
for quasi-median graphs by Bandelt and Mulder and for weakly median
graphs by Bandelt and the speaker. M. Chastand developed a general
theory of fiber-complemented and pre-median graphs for which he showed
that properties similar to (A)-(C) hold provided they hold for prime
graphs (i.e., graphs from the class which cannot be further decomposed
via Cartesian products and gated amalgams). Chastand also asked about
the characterization of prime pre-median graphs.
In this talk, after a brief
introduction to median graphs and their properties (A)-(E), we will
introduce bucolic complexes (a common generalization of
systolic complexes and of CAT(0) cubical complexes) and their
1-skeletons, bucolic graphs. Bucolic complexes are defined as simply
connected prism complexes satisfying some local
combinatorial conditions. We study bucolic complexes from
graph-theoretic and topological perspective. In
particular, we characterize bucolic complexes by some properties of
their 2-skeletons and 1-skeletons. We also show that bucolic
complexes are contractible, and satisfy some
nonpositive-curvature-like properties. Bucolic graphs are particular
pre-median graphs. We show that the prime bucolic
graphs are the weakly bridged graphs (1-skeletons of weakly
systolic complexes), and, using this, we show that bucolic graphs
satisfy properties similar to (A)-(C). In a
subsequent work, we extended the local-to-global characterization of
2-skeletons of bucolic complexes to triangle-square complexes of all
weakly modular graphs and some of their subclasses (Helly graphs,
modular graphs). We also answers Chastand's question by showing that a
pre-median graph is prime if and only its triangle complex is simply connected.
The talk is based on a joint work "Bucolic complexes" with B.
Bostjan, J. Chalopin, T. Gologranc, D. Osajda and on an ongoing
joint work with J. Chalopin, H. Hirai, D. Osajda.
Yves de Cornulier
Titre : Commensurating actions and Property FW
Résumé :
A group has Property FW if
every action on a connected median graph has a finite orbit. This can
be characterized using commensurating actions on sets, and is implied
by Kazhdan's Property T. I'll review various examples, and I'll also
mention a stronger rigidity property, addressing actions on groups
commensurating a subgroup.
Bogdan Nica
Titre : Finiteness at infinity
Résumé :
The boundary of a hyperbolic group enjoys a measure-theoretic finiteness property, called Ahlfors regularity. I will discuss two manifestations of this property:
a hyperbolic group acts properly isometrically on an Lp-space associated to the boundary;
the C*-algebra which encodes the action of a hyperbolic group on its
boundary has all its K-homology representable by finitely summable
cycles.
Damian Osajda
Titre : Infinitely presented graphical small cancellation groups with the Haagerup property
Résumé :
We explore groups defined
by infinite presentations satisfying graphical small cancellation
conditions. We provide natural combinatorial conditions on such
presentations implying the existence of proper actions of the groups on
spaces with walls, and thus the Haagerup property. Consequently, the
groups embed coarsely into Hilbert spaces and the strong Baum-Connes
conjecture holds for them. In particular, we show that infinitely
presented classical C'(1/6) small cancellation groups act on spaces
with walls. We describe further potential applications to the
C*-algebras and the fixed-point properties of the group.
This is a joint work with Goulnara Arzhantseva.
Piotr Przytycki
Titre : Realisation and dismantlability
Résumé :
This is joint work with S. Hensel and D. Osajda. We will discuss "dismantlability" of the arc graph and other graphs in geometric topology. Dismantlability leads to fixed-point theorems for finite group actions. We will explain how to deduce realisation results, in particular Nielsen Realisation Problem for surfaces with nonempty boundary.
Alain Valette
Titre : Irreducible affine isometric actions of groups on Hilbert space.
Résumé :
Affine irreducible
actions were first considered by Y. Neretin (1997), who provided
several examples and proved a remarkable rigidity result: the
restriction of an affine irreducible action, to a co-compact lattice,
is still irreducible. In joint work with B. Bekka and T. Pillon, we
first establish an affine Schur lemma: an affine action is irreducible
iff its commutant (in the monoid of continuous affine transformations)
consists of translations. As consequences, we characterize affine
irreducible actions of nilpotent groups, and we show that, for $G$ a
non-amenable ICC group, the left regular representation is the linear
part of an affine irreducible action, if and only if the first
$L^2$-Betti number of $G$ is at least 1. As an application of our
study, we obtain that, for every locally compact group $G$ containing a
co-compact lattice, the first $L^2$-Betti number of $G$ is bounded
below by the sum, over all square-integrable representations $\sigma$
of $G$, of the products $d_\sigma\times\dim H^1(G,\sigma)$, where
$d_\sigma$ is the formal dimension of $\sigma$.
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