The surface subgroup question asks for the existence of a subgroup of a given group H which is
isomorphic to the fundamental group of a closed surface of genus at least 2. We begin with a survey on
recent progress on this question. We then explain the strategy to construct such surface subgroups in
any cocompact lattice of a classical simple Lie group of non-compact type different from SO(n,1) for
even n. The construction is explicit and geometric and extends earlier work of Kahn and Markovic and of
Kahn, Labourie, Mozes.
Für Kurzentschlossene: Anrechenbarer Vorab-Kurs Mathematik der RWTH Aachen
Die RWTH Aachen bietet in den Semesterferien den Online-Kurs ?HM4Mint-Intensiv? an. Dieser Kurs besteht aus Vorlesungen, Übungen und einer Klausur am Ende des Kurses. Eine erfolgreiche Teilnahme an dem Kurs (inklusive Klausur) kann die Veranstaltung ?Analysis für Studierende der Informatik? des 1. Semesters ersetzen. Für die Anrechnung ist es wichtig die Vertiefung Analysis (3a) und mehrdimensionale Analysis (4a) zu wählen. Für Studierende, die Lust haben schon vor Semesterstart einen Teil der Mathematik vorzuziehen oder auch nachzuholen, könnte dieses Angebot von Interesse sein.
Der Kurs beginnt bereits am 2. August und endet am 2. Oktober mit einer Klausur.
Weitere Informationen sind hier zu finden:
Sollte eine direkte Anmeldung nicht mehr möglich sein, können Sie versuchen die Mitarbeiter*innen des HM4Mint-Intensiv Kurses direkt über email zu erreichen. Die Kontaktdaten finden Sie auf der oben angegebenen Homepage.
Abstract: Approximate groups were identified as a natural framework for geometric group theory by Björklund and Hartnick and further developed by Cordes, Hartnick and Toni?, unifying previous research on apparently disparate areas such as finite approximate groups (Breuillard, Green,
Tao) and quasi-crystals (Meyer and others).
Approximate groups arise naturally via a cut-and-project procedure from lattices in locally compact groups. A central point I want to make is that S-arithmetic groups are, by their standard definition, the result of cut-and-project procedure. They happen to be groups as long as S contains all infinite places, an assumption usually imposed.
In the context of approximate groups, that assumption can be lifted and gives rise to S-arithmetic approximate groups in characteristic 0 that are not groups but resemble S-arithmetic groups in positive characteristic. The finiteness properties of S-arithmetic subgroups of reductive groups in positive characteristic are determined by the Rank Theorem (joint with Bux and Köhl). I will present joint work with Tobias Hartnick proving a Rank Theorem for S-arithmetic approximate groups in characteristic 0.
Abstract: In this talk, I discuss the general question of how to obstruct and construct group actions on manifolds. I will focus on large groups like Homeo(M) and Diff(M) about how they can act on another manifold N. The main result is an orbit classification theorem, which fully classifies possible orbits. I will also talk about some low dimensional applications and open questions. This is joint work with Kathryn Mann.
Abstract: We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions n<=5, a fact that leads us naturally to ask whether this may hold for any n. One consequence of this result is the existence of hyperbolic groups with finite-type subgroups that are not hyperbolic.
The main tool is Bestvina - Brady theory applied to some hyperbolic n-manifolds that decompose very nicely into right-angled polytopes, enriched with the combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.
Abstract: After a brief review of the Arnol'd Conjecture, I will give an overview of the proof of the following joint result with Blumberg: for every closed symplectic manifold, the number of time-1 periodic orbits of a non-degenerate Hamiltonian is bounded below by the rank of the cohomology with coefficients in any field. The case of characteristic 0 was proved by Fukaya and Ono as well as Li and Tian. The new ingredient in our proof is the construction of generalised Floer cohomology groups with coefficients in Morava K-theory. This means that we have to use higher dimensional moduli spaces of pseudo-holomorphic curves, and extract "fundamental chains" in generalised homology.