Past meetings
October 21, 2021: main talk at 2:00 in SR5 at the MI
Annette Karrer (Technion)
Contracting boundaries of right-angled Coxeter groups
Associated to a complete CAT(0) space is a topological space called a contracting or Morse boundary. This boundary indicates how similar the CAT(0) space is to a hyperbolic space. Charney-Sultan proved that this boundary is a quasi-isometric invariant, i.e., it can be defined for CAT(0) groups. In this talk we will study contracting boundaries of right-angled Coxeter groups. Right-angled Coxeter groups are CAT(0) groups defined by graphs. In the main part of the talk, we will study when the contracting boundary of a right-angled Coxeter group with totally disconnected contracting boundary remains totally disconnected if we glue certain graphs on its defining graph. This was part of my dissertation. At the end of the talk, we will use our insights to discuss an interesting example where surprising circles appear in the contracting boundary. This was joint work with Marius Graeber, Nir Lazarovich, and Emily Stark.
October 28, 2021: main talk at 2:00 in SR5 at the MI
Pierre-Louis Blayac (MPIM)
The proximal limit set of divisible convex sets
A divisible convex set is an open subset of a real projective space which is convex and bounded in some affine chart, and on which acts cocompactly a discrete group of projective transformations.
After recalling classical results on this topic, we will explain how one can use a rank-one notion due to M. Islam, and a higher-rank rigidity result dues to A. Zimmer, to answer (positively) the following question of Benoist. Does the proximal limit set of any irreducible non-symmetric divisible convex set coincides with the whole projective boundary of the convex set? We will also recall why answering this enables us to describe the proximal limit set of any divisible convex set.
November 4, 2021: main talk at 2:00 in SR5 at the MI
Feng Zhu (Technion)
Relatively Anosov representations: three characterisations
Convex co-compact subgroups of rank-one Lie groups such as SL(2,R) are a nice class of discrete hyperbolic subgroups with good geometric and dynamical properties. Geometrically finite subgroups of rank-one Lie groups are a slightly larger class of discrete subgroups which allow for certain controlled failures of hyperbolicity but still have relatively good geometric and dynamical properties. Anosov representations generalize the notion of convex co-compactness in rank one to higher-rank semi-simple Lie groups, such as SL(d,R) with d at least 3, giving us an open subset of the space of representations consisting of quasi-isometric embeddings.
Relatively Anosov representations are a common generalization of geometric finiteness in rank one and Anosov representations: they keep some of the good properties of Anosov representations, while allowing for (certain) parabolic elements. We will introduce three characterisations of relatively Anosov representations (based on singular value gaps, flows, and limit maps), with examples.
Parts of this talk represent joint work in progress with Andrew Zimmer.
November 11, 2021: main talk at 5:30 (CET) online
Brian Collier (UC Riverside)
Slodowy slices, Higgs bundles and higher Teichmüller spaces
In this talk I will discuss a Higgs bundle version of the Slodowy slice through a nilpotent orbit in a semisimple Lie algebra. For certain classes of nilpotents related to positivity, this construction describes connected components of the character variety for certain real Lie groups which consist entirely of discrete faithful representations. We will then discuss some nonpositive examples and see what can be said in these cases.
November 18, 2021: main talk at 2:00 in SR5
Rym Smai (U Avignon)
Conformally flat spacetimes with complete photons.
In 2013, Rossi proves that if a maximal globally hyperbolic (abbrev. GHM) conformally flat spacetime has two distinct homotopic lightlike geodesics with the same ends then it is a finite quotient of the Einstein universe. In this case, the ends of such lightlike geodesics are said to be conjugate. In the continuity of this result, I am interested in describing GHM conformally flat spacetimes with complete lightlike geodesics (i.e. which develop as lightlike geodesics joining two conjugate points in the Einstein universe). In this talk, I will describe examples of such spacetimes, that I call Misner domains of the Einstein universe. Under some hypothesis, I prove that the universal covering of a MGH conformally flat spacetime with complete lightlike geodesics contains a Misner strip. The goal would be to prove that any MGH Cauchy compact conformally flat spacetime can be obtained by grafting (or removing) a Misner strip to (from) another one. This can be thought as the Lorentzian analogous of the operation of grafting on hyperbolic surfaces introduced by Thurston.
November 25, 2021
No seminar
December 2, 2021: main talk at 2:30 (CET) online
Pratyush Sarkar (Yale)
Local mixing of diagonal flows on Anosov homogeneous spaces
Strong mixing, such as the results of Babillot and Winter for convex cocompact rank one spaces, have many applications in homogeneous dynamics. In the infinite measure setting, one hopes for local mixing such as the results of Thirion and Sambarino. We will discuss a local mixing result for Anosov homogeneous spaces which generalizes the aforementioned results.
December 9, 2021: main talk at 5:00 (CET) online
Theodore Weisman (UT Austin)
Extended convergence dynamics and relative Anosov representations
Anosov representations are a higher-rank generalization of convex cocompact subgroups of rank-one Lie groups. They are only defined for word-hyperbolic groups, but recently Kapovich-Leeb and Zhu have suggested possible definitions for an Anosov representation of a relatively hyperbolic group - aiming to give a higher-rank generalization of geometrical finiteness.
In this talk, we will introduce a more general version of relative Anosov representation which also interacts well with the theory of convex projective structures. In particular, the definition includes projectively convex cocompact representations of relatively hyperbolic groups, and allows for deformations of cusped convex projective manifolds (including hyperbolic manifolds) in which the cusp groups change in nontrivial ways.
December 16, 2021: main talk at 2:00 in SR5 at the MI
Alessandro Sisto (Heriot-Watt University)
(Hierarchically) hyperbolic quotients of mapping class groups
Thurston's hyperbolic Dehn filling theorem is a way to construct closed hyperbolic 3-manifolds from finite-volume ones. This has a fee group-theoretic analogues, and in the context of Mapping class groups a reasonable analogue of Dehn fillings are quotients by large powers of Dehn twists. I will discuss these and related quotients, which in particular provide many infinite hyperbolic quotients of mapping class groups in low complexity. Based on joint works with Dahmani-Hagen and Behrstock-Hagen-Martin.
January 17, 2022: main talk at 2:00 in SR4 at the MI
Xenia Flamm (ETH)
The real spectrum compactification of the Hitchin component
The space of representations of the fundamental group of a closed surface into PSL(n,R) contains a distinguished connected component: the Hitchin component. The goal of this talk is to introduce the real spectrum compactification of the Hitchin component, echoing the work of Brumfiel in the case of PSL(2,R). We will explain the relevant concepts from real algebraic geometry and provide a characterisation of boundary points in terms of positive limit maps into flag varieties over real closed field extensions of R.
January 20, 2022: main talk at 2:00 in SR5 at the MI
Matteo Migliorini (SNS)
Hyperbolic manifolds fibering over S^1
Fibering over S^1 is a condition, for a hyperbolic manifold, which is rather paradoxical: fibering hyperbolic manifolds can only be found in odd dimensions due to an Euler characteristic constraint, and even then fibers must be very far from being geodesic.
The first example of a fibering hyperbolic 3-manifold was only found in 1977 by Jørgensen; nowadays we have a good grasp of the situation in dimension 3 thanks to Thurston's Geometrisation Theorem, but in higher dimensions almost nothing is known.
After having taken a look at what happens in dimension 3, we will show the existence of a fibering hyperbolic 5-manifold, by using a combinatorial game recently introduced by Jankiewicz, Norin, and Wise, which is in turn based on Bestvina-Brady theory.
January 27, 2022
No Seminar
February 3, 2022
No Seminar
February 10, 2022: main talk at 3:30 (CET) online
Tushar Das (U Wisconsin-La Crosse) - canceled
February 17, 2022
No Seminar