Borel Seminar 2018
Topology and Dynamics in the Swiss Alps

Titles and Abstracts

Mini courses

Aaron Brown

Title: Zimmer's conjecture for actions by cocompact lattices

Abstract: I will present the details of the recent results, due to myself, David Fisher, and Sebastian Hurtado, establishing Zimmer's conjecture for actions by cocompact lattices in SL(n,R): every smooth action on a manifold of dimension at most n-2 is a finite action. If time permits, I will also discuss modifications for lattices in other groups and for non-uniform lattices.

Marc Burger

Title: Introduction to Higher Teichmueller Theory

Abstract: In this minicourse we will report on some recent developments in the area of Higher Teichmueller theory, whose aim is to single out connected components of the G-representation variety of the fundamental group of a compact surface S which are formed of representations with geometric significance.

Lecture I
For G=PSL(2,R) the component of interest, Teichmueller space, is formed by all holonomy representations of hyperbolic structures on S. We'll describe two characterizations of Teichmueller space, one by Fenchel Nielsen coordinates which for G=PSL(n,R) leads to the Hitchin component, and the other by the maximality of the Euler number which for G=Sp(2n,R) leads to the components formed by maximal representations. From the point of view of geometric group theory these representations give in all cases rise to quasi-isometric embeddings and this connection is provided by the concept of Anosov representation.

Lecture II
In this lecture we'll concentrate on maximal representations into Sp(2n,R) and explain the role of bounded cohomology in their study: this allows to extend the notion of maximality to surfaces with boundary, which classical cohomology would not provide, and leads to Fenchel Nielsen coordinates describing components of maximal representations. We will then show how very classical facts from hyperbolic geometry in dimension two, like the collar lemma and the length inequalities obtained by splitting self-intersecting geodesics generalise to this context, while they fail for quasifuchsian representations into SL(2,C).

Lecture III
One of the richest objects in the geometry of surfaces is Thurston's compactification of Teichmueller space;its importance lies, among other things, in the fact that it is a topological ball and that boundary points are related to measured laminations and actions on real trees. In this lecture, following an approach outlined by G.Brumfiel for PSL(2,R), we will show how the structure of real semi algebraic set on the Sp(2n,R)-conjugacy classes of maximal representations leads to a compactification with controlled topology, whose boundary points are linked to representations over non-archimedean real fields, to actions on buildings and to geodesic currents. This compactification is a quite intriguing object even for PSL(2,R) where it strictly dominates Thurston's compactification.

Barbara Schapira

Title: Dynamics of geodesic flows on strongly positively recurrent manifolds and applications

Abstract: I will introduce the notion of strongly positively recurrent (SPR) manifolds. It is a wide class of negatively curved manifolds of infinite volume which, from a dynamical point of view, look like compact manifolds. As an illustration, I will present a result (joint work with S. Tapie) on regularity of topological entropy along smooth variations of the metric of such a SPR manifold. If time allows it, I will present another result (joint work with S. Tapie, R. Coulon, R. Dougall) on the relation between entropy and covers: the topological entropy of a regular cover of a SPR manifold equal the entropy of the SPR manifold iff the cover is amenable.

Talks

Yves Cornulier

Title: Near actions

Abstract: A near action of a group on a set X is defined in the same way as an action, but "modulo indeterminacy on finite subsets of X. We will define this precisely, and will introduce notions allowing to "measure" to which extent a near action is close to "lift" to a genuine action. We will emphasize the following natural occurrence: the near action of the group of piecewise continuous self-transformations of the circle, and that of its subgroup of piecewise isometric self-transformations, better known as "group of interval exchange with flips".

Danijela Damjanovic

Title: Global rigidity for some partially hyperbolic actions

Abstract: In the context of partially hyperbolic actions with compact center foliation, I will present some recent findings on how sufficiently rich Weyl chamber picture of the action forces algebraic structure on the manifold and smooth conjugacy to an algebraic action. This is joint with Disheng Xu and Amie Wilkinson.

Thomas Delzant

Title: Product set growth in groups and hyperbolic geometry

Abstract: Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant α>0 such that for every finite subset U that is not contained in a virtually cyclic subgroup, one has |Un|≥(αU) [(n+1)/2]. Similar estimates are established for groups acting or hyperbolic spaces (join work with M. Steenbock).

Ursula Hamenstädt

Title: Amenable actions and flat cocycles

Abstract: Consider a closed hyperbolic surface S and a representation of the fundamental group of S into a simple Lie group. We show that dynamical properties of the resulting flat bundle over the unit tangent bundle of S, equipped with the Lebesgue measure, coincide with properties known for random walks. We then discuss a large number of other cases where these results are true, e.g. cocycles over geodesic flows on rank one manifolds or the Teichmüller flow.

Fanny Kassel

Title: Convex cocompactness in projective space

Abstract: Anosov representations are representations of Gromov hyperbolic groups into semisimple Lie groups with good dynamical properties; they play an important role in higher Teichmüller theory. We will relate this notion to a notion of convex cocompactness in projective space, and discuss generalizations and examples. Joint work with J. Danciger and F. Guéritaud.

Dieter Kotschick

Title: Lagrangian foliations and Anosov symplectomorphisms on Kaehler manifolds

Abstract: We investigate parallel Lagrangian foliations on Kaehler manifolds. On the one hand, we show that a Kaehler metric admitting a parallel Lagrangian foliation must be flat. On the other hand, we give many examples of parallel Lagrangian foliations on closed flat Kaehler manifolds which are not tori. These examples arise as the (un-)stable foliations of Anosov automorphisms preserving a Kaehler form. This is joint work with M.J.D. Hamilton.

Jean-François Lafont

Title: Finiteness of maximal geodesic submanifolds in hyperbolic hybrids

Abstract: I will give an overview of the various constructions of hyperbolic manifolds. I will then focus on the hyperbolic hybrids, a class of non-arithmetic hyperbolic manifolds. I will explain why these manifolds contain only finitely many maximal totally geodesic closed (immersed) submanifolds. This is joint work with Fisher, Miller, and Stover, and is the first positive result towards a question raised independently by Alan Reid and Curt McMullen. This talk will be targeted at a general audience.