MedianOrsayTitresResumes

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.

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.

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.

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 $\sigma$.