# Schedule

Here is a tentative schedule. Registration from 1:30 to 2 on Monday 24. Talks are one hour long (conference room, second floor) with 5 minutes in between  talks. Coffee breaks are scheduled at 10 a.m. and 4:10 p.m. and take place in Döblin room at the end of the fourth floor.

Time Monday Tuesday Wednesday Thursday Friday
9 Caprace Parreau Karlsson Cornulier
10:30 Cornulier Caprace Parreau Karlsson
11:40 Karlsson Cornulier Caprace Parreau
12:40 Registration from 1:30 Lunch Lunch Lunch Lunch
2 Valette Iozzi Leeb Pansu
3:10 Wenger Lamy Lytchak Navas
4:40 Le Boudec Lécureux Py
7:30 Conference Dinner

## Mini-courses

#### Pierre-Emmanuel Caprace

From amenability to buildings in non-positive curvature

The goal of these lectures, based on joint work with Nicolas Monod, is to describe the proper CAT(0) admitting a cocompact action of an amenable group. The starting point is a theorem of Adams and Ballmann, implying that the space is (quasi)flat if the group action is properly discontinuous. I will explain how removing the latter condition leads to connections with spherical and Euclidean buildings, non-positively curved Lie groups, and Gromov hyperbolic spaces with a homogeneous boundary.

#### Yves Cornulier

Large-scale geometry of Lie groups

The study of connected Lie groups up to quasi-isometry is very rich. I will introduce various quasi-isometry invariants of metric space in order to investigate them in the case of connected Lie groups. One of these invariants is the asymptotic cone of a metric space, which is another metric space purporting to describe what we can see of the original space when viewed from infinitely far away. I will describe various examples, and also explain how the study boils down to the study in the case of simply connected solvable Lie groups. I will also mention some more invariants, including asymptotic dimension, and address the issue of being non-positively curved at large scale, which, unlike Gromov-hyperbolicity, is not well-understood in this context.

#### Anders Karlsson

Nonpositive curvature, metric functionals and ergodic theorems

Given a metric space one defines its metric functionals. For proper spaces these are commonly called horofunctions. For CAT(0) spaces one has a well-known “representation theorem” that horofunctions are in correspondence, via Busemann, with geodesic rays. This is similar to the theory of normed vector spaces and their continuous linear functionals. There is a general spectral principle in the metric setting which in specific situations leads to results like the von Neumann-Carleman ergodic theorem, the Wolff-Denjoy theorem in complex dynamics, a spectral theorem for surface homeomorphisms due to Thurston as well as new results. An advantage of the metric setting compared with the various arguments in the specific settings is that it extends to ergodic cocycles which leads to multiplicative ergodic theorems (joint work with S. Gouëzel extending en earlier joint work with F. Ledrappier). In one such new result for bounded invertible operators, the infinite dimensional symmetric space of positive operators with its weak form of nonpositive curvature plays an important role.

#### Anne Parreau

Introduction to real Euclidean buildings.

Real Euclidean buildings are CAT(0) metric spaces with many flats and a rich combinatorial structure.  They may be seen as higher rank generalizations of real trees, or ultrametric analoguous of Riemannian symmetric spaces.  In my lectures I will give a short introduction to real Euclidean buildings, concentrating on the example of $\operatorname{SL}_n(\mathbf{K})$, and discuss some of their geometric properties, in particular the boundary at infinity, the relationships with the projective geometry, the properties of the refined distance with value in a model flat, and weak convexity.

## Research talks

#### Alessandra Iozzi

Bounded cohomology, boundary maps, and the Roller boundary

We recall the construction of the Roller boundary of a (not necessarily locally finite) CAT(0) cube complex and show how its properties lead to rigidity results using the median class of an action.  In the case of a tree we illustrate how this can be used to construct median quasimorphisms and characterize properties of an action.

#### Stéphane Lamy

The Cremona group acting on an infinite dimensional hyperbolic space

The Cremona group is the group of birational self-maps of the plane.
This group comes from the algebro-geometric world, but it shares many features with groups such as the modular group $\operatorname{SL}(2,\mathbf{Z})$, or the mapping class group of a Riemann surface.
I will describe how one can make the Cremona group acts on an infinite dimensional hyperbolic space, and some of the applications : Tits alternative, non simplicity…
I will also describe the link with the notion of dynamical degree, which is an analogue to the stretch factor in the context of a pseudo-Anosov map in the mapping class group.

Groups acting on trees with almost prescribed local action.

Burger-Mozes’ groups are groups acting on a regular tree, and whose local action is prescribed by a permutation group. This talk deals with a family of groups defined by relaxing this local rigidity condition. These groups are connected to Neretin’s group of almost automorphism of a tree.

We will explain how this construction yields examples of “small” simple compactly generated locally compact groups, and we will address the study of their lattices.

#### Jean Lécureux

Non-linearity of groups acting on exotic affine buildings

The affine buildings of type $\operatorname{A}_2$ are most useful in the study of $\operatorname{PGL}_3(\mathbf{k})$ and its subgroup, where k is a local field. However, there are many more such buildings, and some of them admit a discrete, cocompact automorphism group. We prove that these groups do not have any infinite representation in any linear group. The techniques used are inspired from a proof of Margulis Superrigidity by Bader and Furman, and an essential ingredient is the ergodicity of some geodesic flow on the building. This is joint work with P.-E. Caprace ane U. Bader.

#### Bernhard Leeb

Finsler compactifications of symmetric and locally symmetric spaces

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant “polyhedral” Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds with corners, of locally symmetric spaces which are orbifold quotients $X/\Gamma$ by arbitrary uniformly weakly regular subgroups $\Gamma<G$. These bordifications result from attaching $\Gamma$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of weakly regular conical antipodal (=$\tau_{mod}$-RCA) subgroups, equivalently, Anosov subgroups. We show, conversely, that $\tau_{mod}$-RCA subgroups are characterized by the existence of such compactifications. This is joint work with Misha Kapovich.

#### Alexander Lytchak

Isoperimetric characterization of non-positive curvature

In the talk I will discuss the following joint result with S. Wenger: A locally compact geodesic metric space is CAT(0) if and only if it satisfies the 2-dimensional Euclidean isoperimetric inequality, hence if and only if any closed curve of length $L$ bounds some disc of area at most $L^2/4\pi$.

The main steps in the proof are the solution of the classical Plateau problem in general spaces and the analysis of the intrinsic structure of area minimizing discs. The result also applies to non-zero upper bounds on curvature.

#### Andrés Navas

A canonical berycenter map on Buseman spaces.

In this talk I will report on a construction of a barycenter for measures with finite first moment on general Buseman spaces, which is 1-Lipschitz with respect to the 1-Kantorovich-Wasserstein metric.

#### Pierre Pansu

The quasi-symmetric Hölder equivalence problem

What is the optimal pinching of curvature on spaces quasiisometric to complex hyperbolic spaces ? This leads to the following problem: what is the best Hölder continuity exponent for a homeomorphism of Euclidean space to a metric space quasisymmetric to Heisenberg group, when the inverse map is assumed to be Lipschitz ? We give a partial result on this question.

#### Pierre Py

Actions of $\operatorname{PO}(n,1)$ on infinite dimensional symmetric spaces

I will recall some classical facts from representation theory which allow to build isometric actions of $\operatorname{PSL}(2,\mathbf{R})$ and more generally of $\operatorname{PO}(n,1)$ on certain infinite dimensional symmetric spaces. Along the way, I will survey some results obtained in collaboration with Thomas Delzant and Nicolas Monod and present a few open questions.

#### Alain Valette

A survey of the Haagerup property

A locally compact group has the Haagerup property (or: is a-$(T)$-menable) if it admits a proper affine isometric action on a Hilbert space. The Haagerup property is a weak form of amenability: the class of Haagerup groups contains amenable groups, but also free groups, Coxeter groups, closed subgroups of $\operatorname{SO}(n,1)$ and $\operatorname{SU}(n,1)$, etc… We will survey the following topics:

1.  Unitary representations vs affine isometric actions.
2.  Geometric characterizations.
3.  Permanence properties.
4.  Applications

#### Stefan Wenger

Plateau’s problem, isoperimetric inequalities, and large scale geometry

In the first part of the talk I will describe a generalization of the classical problem of Plateau to the setting of metric spaces. I will show that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space admits a local quadratic isoperimetric inequality for curves then such a disc is locally Hölder continuous in the interior and continuous up to the boundary. In the second part of the talk I will explain how existence and regularity of area minimizers can be used to study the local and the large scale geometry of spaces with a quadratic isoperimetric inequality for curves. The talk is based on joint work with Alexander Lytchak.