Heidelberg Differential Geometry Seminar

Winter 2018/19

Next upcoming talk:

Thu Feb 14  ∙  Seminarraum 3  ∙  13:30

Leon Carvajales (Paris)    Anosov representations and counting in some PSO(p,q)-symmetric spaces

For positive integers p and q consider a quadratic form on R^{p+q} of signature (p,q) and let O(p,q) be its group of linear isometries. The space X of q-dimensional subspaces of R^{p+q} on which the quadratic form is negative definite is the Riemannian symmetric space of PSO(p,q). Let S be a totally geodesic copy of the Riemannian symmetric space of PSO(p,q-1) inside X. We look at the orbit of S under the action of a projective Anosov subgroup of PSO(p,q). For certain choices of such a subgroup we show that the number of points in this orbit which are at distance at most t from S is asymptotically purely exponential as t goes to infinity. We also provide a geometric interpretation of this result in the pseudo-Riemannian hyperbolic space of signature (p,q-1).

Fri Mar 1  ∙  Seminarraum 3  ∙  11:00

Rene Ruehr (Tel Aviv)    Counting saddle connections in a homology class modulo q

We give effective estimates for the number of saddle connections on a translation surface that have length less or equal than L and are in a prescribed homology class modulo q. Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum. This is joint work with Michael Magee, and builds on recent advances by Magee on Selberg's eigenvalue conjecture for moduli spaces of abelian differentials, and Rodolfo Gutierrez-Romo's work on Zorich's conjecture on Zariski-density of Rauzy-Veech groups.

Past talks:

Thu Feb 7  ∙  Seminarraum 3  ∙  12:45

Nguyen-Thi Dang (IRMA Rennes)    Topological mixing of the Weyl chamber flow

I will talk about a joint work with O. Glorieux. Let $G$ be a semisimple, real linear, connected Lie group and $\Gamma$ a Zariski dense, discrete subgroup of $G$. Let $A$ be a maximal torus, $K$ be a maximal compact subgroup and $M$ be the centralizer of $A$ in $K$. When $G=\mathrm{PSL}(2,\mathbb{R})$, the right action by multiplication of $A$ on $\Gamma\backslash G$ identifies with the action of the geodesic flow on $T^1 \Gamma \backslash \mathbb{H}^2$. It is well known that this flow is topologically mixing on its non-wandering set. When $\Gamma \backslash G/M$ is non compact of infinite volume and $G$ of higher real rank, what can we say about the right action of one parameter subgroups of $A$ acting by right multiplication on $\Gamma \backslash G/M$? First, I will give an example in higher rank of space of Weyl chambers and introduce in this situation the Weyl chamber flows I study. I will then state our result with Olivier Glorieux, a necessary and sufficient condition for topological mixing. Basing myself on the previous example, I will explain the geometric situation in higher rank. Lastly, I will give the main ideas behind our proof.

Thu Dec 13  ∙  Seminarraum 3  ∙  12:45

Elia Fioravanti (Oxford)    Superrigidity of actions on finite rank median spaces

Finite rank median spaces simultaneously generalise real trees and finite dimensional CAT(0) cube complexes. Requiring a group to act on a finite rank median space is in general much more restrictive than only asking for an action on a general median space. This is particularly evident for certain irreducible lattices in products of rank-one simple Lie groups: they admit proper cocompact actions on infinite rank median spaces, but any action on a (complete, connected) finite rank median space must fix a point. Our proof of the latter fact is based on a generalisation of a superrigidity result of Chatterji-Fernós-Iozzi. We will sketch the necessary techniques, focussing on some intriguing similarities between horofunction compactifications of median spaces and Satake compactifications of symmetric spaces.

Thu Nov 29  ∙  Seminarraum 3  ∙  12:45

Tam Nguyen-Phan (MPI Bonn)    Topology of ends of nonpositively curved manifolds

Locally symmetric manifolds of noncompact type form an interesting class of nonpositively curved manifolds. The topology of the end of an arithmetic locally symmetric space is controlled by an arithmetically-constructed object called the "rational Tits building". The rational Tits building can be thought of abstractly or as a subset of the visual boundary of the universal cover of M and is homotopically a wedge of spheres of dimension q-1, where q is the "Q-rank" of the locally symmetric space. In general, q is less than or equal to half the dimension of the locally symmetric space. We show that this is not an arithmetic phenomenon but a consequence of nonpositive curvature alone. We build a geometric analog of the rational Tits building for general noncompact, finite volume, complete, n-manifolds of bounded nonpositive curvature. We use this to show that any polyhedron, in the thin part (i.e. the end) of M that lifts to the universal cover can be homotoped within the thin part of M to one with dimension less than or equal to (n/2 - 1). Loosely speaking, this says that any topological feature that survives from being pushed to infinity must be in dimension less than n/2. I will describe how this is done. This is joint work with Grigori Avramidi. This talk is about nonpositively curved geometry. No knowledge of Tits buildings is required (or will be given).

Thu Nov 22  ∙  Seminarraum 3  ∙  12:45

Carlos De la Cruz Mengual (ETH Zürich)    Stability in bounded cohomology for classical groups

Homological stability is a phenomenon that has been studied and established in the context of ordinary group homology for several infinite ascending series of groups. So far the only existing stability result known for bounded cohomology, by Monod, concerned the families of the general and special linear groups over any local field. In this talk we present an argument that proves stability for the symplectic families over the fields of real and of complex numbers. We first describe a general method that guarantees bounded-cohomological stability along a series of locally compact second-countable groups, provided that there exists a family of highly connected complexes on which the groups have a highly transitive action. Then, we introduce a new family of complexes associated to the symplectic groups, which we call symplectic Stiefel complexes. Similar kinds of objects can be defined for other families of classical groups. This is joint work with Tobias Hartnick.

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