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Tee-Seminar der AG KramerZeit und Ort: ZOOM
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Inhalt:
Mitglieder der Arbeitsgruppe und Gäste tragen über ihre laufenden Forschungsarbeiten vor, oder über Themen, die uns interessieren.
Vorträge:
- 13.04.2021 Giles Gardam (Münster) Kaplansky's conjectures
- 20.04.2021 Linus Kramer (Münster) Proper actions and buildings
- 27.04.2021 Jason Behrstock (CUNY) Hierarchically hyperbolic groups: an introduction
- 04.05.2021 Rafael Dahmen (KIT) Direct limits of topological groups
- 11.05.2021 Olga Varghese (Magdeburg) Automatic continuity for groups whose torsion subgroups are small
- 18.05.2021 Jonas Beyrer (IHES) Aspects of higher rank Teichmüller theory
- 01.06.2021 Federico Vigolo (Münster) Cube complexes with coupled links
- 15.06.2021 Ivan Levcovitz (Technion) Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques
- 22.06.2021 Yuri Santos Rego (Magdeburg) Recognizing graphs in three-space
- 06.07.2021 Wolfgang Pitsch (Universitat Autònoma de Barcelona) The Maslov index from the viewpoint of Sturm sequences
- 13.07.2021 Emily Stark (Wesleyan University) Action rigidity for graphs of manifold groups.
Abstract: Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other conjectures and group properties, and finish with my recent counterexample to the unit conjecture.
Abstract: I will discuss some useful facts about proper actions on proper metric spaces. One application is that proper, Weyl-transitive actions on locally finite buildings are line-transitive.
Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmüller space, most cubulated groups, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view, both describing new tools to use to study these groups and applications of those results. This talk will include joint work with Mark Hagen and Alessandro Sisto.
Abstract: Given a directed system of topological groups, one can consider the direct limit (colimit) in the category of topological spaces. Unfortunately, sometimes this topology may fail to be a group topology due to discontinuity of the multiplication map. In these cases the topology underlying the colimit in the category of topological groups is different from the colimit in the category of topological spaces. In this talk, I want to present some well-known results on when this pathology occurs in the case of countable directed systems - as well as some newer results on certain uncountable systems (called "long directed systems") which behave very differently than countable ones. This will be illustrated by some (hopefully) motivating examples. This is joint work with Gábor Lukács.
Abstract: In the category of locally compact Hausdorff groups LCG one has to distinguish between algebraic morphisms and algebraic and continuous morphisms. Let Epi(L,G) be the set of surjective group homomorphisms and cEpi(L,G) the subset consisting of continuous surjective group homomorphisms. The question we address is the following: Under which conditions on the discrete group G does the equality Epi(LCG,G)=cEpi(LCG,G) hold?
Abstract: 'Classical' Teichmüller space is the space of hyperbolic structures on a closed surface S. If H is the fundamental group of S, Teichmüller space can also be identified with a connected component of Hom(H,PSL(2,R))/PSL(2,R) that consists entirely of discrete and faithful representations. Surprisingly there are other Lie groups G (of higher rank) such that there exist also connected components of Hom(H,G)/G consisting entirely of discrete and faithful representations - those components are called 'Higher rank Teichmüller spaces'. In the last two decades there has been a lot of research studying those spaces, in particular the similarities (and differences) to classical Teichmüller space. In this talk I try to give a short introduction and overview of Higher rank Teichmüller theory and then discuss joint work with B. Pozzetti on this topic.
Abstract: In this talk I will introduce a procedure to construct examples of non-positively curved cube complexes. The construction we suggest takes as input two finite simplicial complexes and gives as output a finite cube complex whose local geometry can be easily described. This local information can then be used to obtain global information, e.g. about cohomogical dimension and hyperbolicity of the fundamental group of the cube complex. This is joint work with Krobert Ropholler.
Abstract: There are few available techniques for constructing non-quasiconvex subgroups of hyperbolic groups. In this talk, I will discuss joint work with Pallavi Dani that provides such a construction by utilizing techniques inspired by Stallings' foldings. The hyperbolic groups we construct are in the natural class of right-angled Coxeter groups and can be chosen to be 2-dimensional. Additionally, the non-quasiconvex subgroups can be chosen to satisfy several additional desirable properties. Explicit examples of such non-quasiconvex subgroups will be outlined in the talk.
Abstract: Spatial graphs - i.e., topological graphs together with an embedding in 3-space (or in the 3-sphere) - generalize knots and links, thus being objects of general interest for multiple reasons. In this talk we will discuss the problem of algorithmically recognizing spatial graphs (and present a solution!) while addressing similarities and differences with the case of knots, links, and associated groups. Based on joint work with Stefan Friedl, Lars Munser and José Pedro Quintanilha.
Abstract: It is a classical fact that Wall's index of three Lagrangians in a symplectic space over a field k defines a 2-cocycle M on the associated symplectic group with values into the Witt group of k. Moreover, modulo the square of the fundamental ideal this is a trivial 2-cocycle. In this talk I will revisit this fact from the viewpoint of the theory of Sturm sequences and Sylvester matrices developed by J. Barge and J. Lannes. This allows us in particular to give an explicit formula for the coboundary associated to the mod I2 reduction of M which is valid on any field of characteristic different from 2.
Abstract: Rigidity theorems prove that a group's geometry determines its algebra, typically up to abstract commensurability. A group's geometry is studied via its quasi-isometry class and via its geometric actions on proper geodesic metric spaces. These two points of view lead to two distinct notions of rigidity: quasi-isometric rigidity and action rigidity, respectively. We will discuss the differences between these two notions of rigidity, focusing on graphs of hyperbolic manifold groups. Applications include new examples of quasi-isometric groups that do not act geometrically on the same proper geodesic metric space. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.