# Courses and Talks

MINI-COURSES (4 hours each)

MINI-COURSES

**- On the structure of group/actions on CAT(0) spaces.
Pierre-Emmanuel Caprace, Univ. Catholique de Louvain.**

The goal of this course is to provide an introduction to the geometry of CAT(0) spaces and the properties of their isometry groups. Starting with a description of prominent families of examples, namely symmetric spaces and Euclidean buildings, we will develop the basics of a theory showing that proper CAT(0) spaces are subjected, at a high level of generality, to rigidity phenomena similar to the familiar properties of those basic examples. The main tool in establishing those results is the study of the full isometry group, which is naturally endowed with a locally compact group topology. Building upon some general considerations on the possibly non-discrete full isometry group, we will illustrate this approach by deducing various properties of discrete groups of isometries: existence of free abelian subgroups, algebraic structure of amenable groups, rigidity of lattices.

The course plan is the following:

- Definitions and key examples

- Geometric density

- The full isometry group

- Applications to discrete subgroups.

The goal of this course is to provide an introduction to the geometry of CAT(0) spaces and the properties of their isometry groups. Starting with a description of prominent families of examples, namely symmetric spaces and Euclidean buildings, we will develop the basics of a theory showing that proper CAT(0) spaces are subjected, at a high level of generality, to rigidity phenomena similar to the familiar properties of those basic examples. The main tool in establishing those results is the study of the full isometry group, which is naturally endowed with a locally compact group topology. Building upon some general considerations on the possibly non-discrete full isometry group, we will illustrate this approach by deducing various properties of discrete groups of isometries: existence of free abelian subgroups, algebraic structure of amenable groups, rigidity of lattices.

The course plan is the following:

- Definitions and key examples

- Geometric density

- The full isometry group

- Applications to discrete subgroups.

**- A crash course on SL(2,R) and Schrödinger cocycles.**

**Raphaël Krikorian, University Pierre et Marie Curie (Paris 6)**

*The aim of the minicourse is to study the spectral properties of quasiperiodic 1D Schrödinger operators. An important tool in this approach is the study of the dynamics of the related Schrödinger cocycles. This approach proved to be very useful these last 20 years.*

*Program: *

*1) Spectral theory of bounded and symmetric operators, Schrödinger operators, Berezansky´s theorem, dynamically defined Schrödinger operators, spectral measures and the integrated density of states.*

*2) Schrödinger cocycles, facts from ergodic theory, rotation number and Lyapunov exponents, m-functions, uniform and non-uniform hyperbolicity, Oseledec theorem.*

*3) Links between the spectral and dynamical aspects, spectrum/nonuniform hyperbolicity, density of states/rotation number, Thouless formula.*

*4) Reducibility of qp cocycles, KAM theory, Dinaburg-Sinai and Eliasson´s reducibility theorems. Links with the absolutely continuous spectrum.*

*5) Anderson localization and its link with nonuniform hyperbolicity.*

*6) Aubry-André duality, application to the Almost-Mathieu operator.*

*Prerequisites: **Hilbert space, basics of complex analysis (holomorphic, harmonic and subharmonic functions), Fourier analysis, measure theory and ergodicity.*

*Bibliography:*

*R. Carmona, J. Lacroix}. Spectral theory of random Schrödinger operators. Birkhauser*

Th. Ransford. Potential theory in the complex plane. LMS

*P. Walters. An introduction to ergodic theory. Springer*

** - Livsic theory and cocycles, including cocycles of diffeomorphisms.
Rafael de la Llave, Georgia Inst. of Technology.
**

The central problem will be the problem of triviality of a cocycle over an Anosov system.

More precisely, we consider an Anosov system f in a manifold M and a mapping phi: M ---> G, where G is a group and we consider the problem of whether one can find another mapping eta:

The central problem will be the problem of triviality of a cocycle over an Anosov system.

More precisely, we consider an Anosov system f in a manifold M and a mapping phi: M ---> G, where G is a group and we consider the problem of whether one can find another mapping eta:

Clearly, there are obstrutions. If p is periodic of period N, we obtain that it is necessary that

phi(p) phi(f(p)) ... phi(f^{N-1}(p)) = Id.

Remarkably, when f is an Anosov diffeomorphism, phi is Hölder and G is a finite dimensional group, this is the only obstruction (the case of commutative, compact or nilpotent groups was obtained by Livsic in the early 70´s; the general case was obtained by Kalinin in 2010).

We plan to explore several ideas around these topics:

1) Motivation. Origins of the problem

2) The basic Livsic argument

3) Regularity of solutions in the finite dimensional case

4) Some variations: conformal methods, hyperbolic sets.

5) The slow variation argument.

6) Infinite dimensional problems.

*M ---> G*in such a way that eta(x) = phi(x) eta(f x).Clearly, there are obstrutions. If p is periodic of period N, we obtain that it is necessary that

phi(p) phi(f(p)) ... phi(f^{N-1}(p)) = Id.

Remarkably, when f is an Anosov diffeomorphism, phi is Hölder and G is a finite dimensional group, this is the only obstruction (the case of commutative, compact or nilpotent groups was obtained by Livsic in the early 70´s; the general case was obtained by Kalinin in 2010).

We plan to explore several ideas around these topics:

1) Motivation. Origins of the problem

2) The basic Livsic argument

3) Regularity of solutions in the finite dimensional case

4) Some variations: conformal methods, hyperbolic sets.

5) The slow variation argument.

6) Infinite dimensional problems.

**- Affine isometric actions.**

**Alain Valette, Univ. de Neuchâtel.**

We will define affine isometric actions on Hilbert spaces (together with the relevant mild cohomological formalism) and give examples from geometry. We plan to give a proof of the following results:

- a group is non-amenable if and only if, whenever an action with linear part the regular representation almost has fixed points, it has fixed points (Guichardet);

- every amenable group admits a proper action on a Hilbert space;

- if an amenable group in Shalom´s class (HFD) (which contains polycyclic groups) embeds quasi-isometrically into Hilbert space, then it is virtually abelian.

We will define affine isometric actions on Hilbert spaces (together with the relevant mild cohomological formalism) and give examples from geometry. We plan to give a proof of the following results:

- a group is non-amenable if and only if, whenever an action with linear part the regular representation almost has fixed points, it has fixed points (Guichardet);

- every amenable group admits a proper action on a Hilbert space;

- if an amenable group in Shalom´s class (HFD) (which contains polycyclic groups) embeds quasi-isometrically into Hilbert space, then it is virtually abelian.

**TALKS (45 min each)**

**- Denseness of domination.
Jairo Bochi, PUC-Rio de Janeiro.**

*Among linear cocycles (vector bundle automorphisms), the projectively hyperbolic ones (i.e., those that have a dominated splitting) form an important subclass.*

Assume that the base dynamics is minimal, and that the fiber dimension is at least 3. Then I prove the following result: for any homotopy class C of cocycles, the (open) subset of C formed by the cocycles that have a dominated splitting is either dense in C or empty. In other words, obstructions to domination are purely topological, in the sense that they cannot be removed by deforming the cocycle.

As I will explain, the proof has two parts: the first part is about expansion rates (Lyapunov exponents), and the second part is about rotations. Each part reduces to a problem of finding almost-invariant sections for a certain skew-product dynamics which is isometric on the fibers. In the first part, the fibers are noncompact and of nonpositive curvature, and the problem is solved cleanly using geometrical tools developed with Navas. In the second part, however, the fibers are compact and have positive curvature. In this case, the construction of almost- invariant sections is more elaborate and ultimately relies on a little-known result in quantitative homotopy theory.

If time allows, I will also explain why the corresponding 2-dimensional statement is false, and how to correct it.

Assume that the base dynamics is minimal, and that the fiber dimension is at least 3. Then I prove the following result: for any homotopy class C of cocycles, the (open) subset of C formed by the cocycles that have a dominated splitting is either dense in C or empty. In other words, obstructions to domination are purely topological, in the sense that they cannot be removed by deforming the cocycle.

As I will explain, the proof has two parts: the first part is about expansion rates (Lyapunov exponents), and the second part is about rotations. Each part reduces to a problem of finding almost-invariant sections for a certain skew-product dynamics which is isometric on the fibers. In the first part, the fibers are noncompact and of nonpositive curvature, and the problem is solved cleanly using geometrical tools developed with Navas. In the second part, however, the fibers are compact and have positive curvature. In this case, the construction of almost- invariant sections is more elaborate and ultimately relies on a little-known result in quantitative homotopy theory.

If time allows, I will also explain why the corresponding 2-dimensional statement is false, and how to correct it.

**- Attempts at nonlinear versions of spectral theory.**

** Anders Karlsson, Univ Geneva.**

*I will describe some statements and questions on spectral aspects of transformations such as diffeomorphisms of compact manifolds and bounded linear operators. Instead of linear spaces, we will study induced actions on nonlinear ones (like symmetric spaces). Two results that provide motivation are Thurston´s spectral theorem for surface homeomorphisms and Oseledets´ multiplicative ergodic theorem for cocycles of matrices.*

** ****- Regularity of the stochastic entropy****.
**

**François Ledrappier, Univ. Notre-Dame**

**.**

We consider a closed negatively curved manifold $(M,g)$ and the stochastic (or Kaimanovich) entropy $h(M,g)$. In this talk, we recall the relations with the other growth rates (volume entropy, bottom of the spectrum). We discuss the $C^1$ regularity of $h(M, g) $ along conformal variations of $g$ and applications. This is a joint work with Lin Shu (Peking University).

We consider a closed negatively curved manifold $(M,g)$ and the stochastic (or Kaimanovich) entropy $h(M,g)$. In this talk, we recall the relations with the other growth rates (volume entropy, bottom of the spectrum). We discuss the $C^1$ regularity of $h(M, g) $ along conformal variations of $g$ and applications. This is a joint work with Lin Shu (Peking University).

**- Stable transitivity of Heisenberg group extensions of hyperbolic systems.**

Viorel Nitica, West Chester Univ.

Viorel Nitica, West Chester Univ.

(Joint work with A.Torok) We show that among C^r extensions (r > 0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group H

(Joint work with A.Torok) We show that among C^r extensions (r > 0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group H

_{n}of dimension 2n+1 that avoid an obvious obstruction, those that are topologically transitive are open and dense.

**-**

**Fixed point property for groups acting on simplicial complexes**

*.*

**Izhar Oppenheim, The Ohio State Univ.**

*Given a group G and a metric space Y, one can ask when does the group G have a fixed point property with respect to Y, (i.e. does every isometric action of G on Y have a fixed point?). When the group G also acts geometrically on a simplicial complex X, one can sometimes prove that G has a fixed point property by "comparing" the local structure of X with the metric structure of Y. The classical example of such theorem is the "Zuk criterion" stating that G has a fixed point property with respect to every Hilbert space if the Laplacian on links of X has a large enough spectral gap. In my talk I want to discuss "generalized Zuk criterion" - i.e. given a metric space Y with certain nice properties (e.g. a reflexive Banach space or a Busemann NPC, uniformly convex metric space), I´ll present a suitable fixed point criterion relying on the interplay between Y and the local structure of X.*

**- An exotic deformation of the hyperbolic space.
Pierre Py, Univ. de Strasbourg.**

I will explain how to construct a continuous family of ´´exotic´´ locally compact CAT(−1) spaces with cocompact isometry groups all isomorphic to the isometry group of the real hyperbolic space H^n. This is joint work with Nicolas Monod.

I will explain how to construct a continuous family of ´´exotic´´ locally compact CAT(−1) spaces with cocompact isometry groups all isomorphic to the isometry group of the real hyperbolic space H^n. This is joint work with Nicolas Monod.

**-**

**A problem involving higher-degree cocycles.**

Felipe A. Ramírez,

Felipe A. Ramírez,

**Univ. of Bristol**.

*I will discuss higher-degree smooth cohomology of higher-rank abelian actions, with emphasis on a conjecture of A. and S. Katok generalizing the Livshitz Theorem to Anosov actions of higher-rank abelian groups. Specifically, the conjecture predicts that the obstructions to solving the top-degree coboundary equation are only those coming from integration over closed orbits (like in the Livshitz Theorem) and that all of the lower cohomologies trivialize (as they are known to do in degree one, by work of A. Katok and R. Spatzier).*

I will outline a representation-theoretic approach to this problem that yields results for certain systems, for example some Weyl chamber flows and also some unipotent actions.

I will outline a representation-theoretic approach to this problem that yields results for certain systems, for example some Weyl chamber flows and also some unipotent actions.

**- Random walks on symmetric groups.**

Andrzej Zuk, Univ. Paris 7.

Andrzej Zuk, Univ. Paris 7.