Oberseminar Geometrie und Topologie

(run by Sebastian Hensel, Christian Lange, and Bernhard Leeb)

The seminar takes place on Thursdays, 16:15, virtually via zoom.
If you want to participate, please send a quick email to Sebastian Hensel to be added to the mailing list, and you will receive invitations and zoom links.

• May 6th, 2021: Richard Webb (Manchester): How non-positively curved is the mapping class group?
  Abstract: In general, mapping class groups are not CAT(0) but one is still interested in finding non-proper yet cocompact actions on CAT(0) spaces. We will show that, in general, the arc complex admits no CAT(0) metric with finitely many shapes. In particular there is no finite-index subgroup of the mapping class group that preserves a CAT(0) metric on the arc complex. The hex theorem from combinatorics plays a role in the proof. The analogous statements are true for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. I will give background and motivation, and some connections of the work with other areas of algebra and topology.
• May 27th, 2021: Alessandro Sisto (Heriot-Watt): What does a generic 3-manifold look like?
  Abstract: I will discuss two constructions of "random" 3-manifolds, namely Heegaard splittings and mapping tori where the gluing map is chosen using a random walk on a mapping class group. As it turns out, in both cases one obtains hyperbolic manifolds with asymptotic probability 1. I will give a brief overview of further geometric properties of these manifolds, and then focus on a deterministic result on Heegaard splittings, joint with Peter Feller and Gabriele Viaggi, that applies to random Heegaard splittings.
• June 10th, 2021: Jose Andres Rodriguez Migueles (LMU) Volumes associated to periodic orbits of the geodesic flow on the Modular surface
  Abstract: Closed geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle. The isotopy class of any periodic orbit can be considered as a knot in a 3-manifold. The complement of those knots is always a hyperbolic 3-manifold, and hence has a well-defined volume. We show that there exist sequences of closed geodesics for which this volume is bounded linearly in terms of the period of the geodesic’s continued fraction expansion.
• July 8th, 2021: Christian Lange (LMU) Closed geodesics on orbifolds
  Abstract: After recalling some basics about orbifolds, we will discuss results on the existence of closed geodesics on orbifolds as well as properties of orbifolds with all geodesics closed. The latter are related to Finsler and systolic geometry as we will see in the second part of the talk.
• July 15th, 2021: Elia Fioravanti (MPI Bonn) Coarse-median preserving automorphisms
  Abstract: We study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups (RAAGs and RACGs). If Phi is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we show that Fix(Phi) is finitely generated and undistorted. Up to replacing Phi with a power, the fixed subgroup is actually quasi-convex with respect to the standard word metric (which implies that it is separable and a virtual retract, by work of Haglund and Wise). Our techniques also apply to automorphisms of hyperbolic groups and to certain automorphisms of hierarchically hyperbolic groups. Based on arXiv:2101.04415.