Tea Seminar of our groupPlace and time:
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Inhalt:
In this seminar, guests and members of our group present their research, or topics of interest to us. Presently we use a hybrid format for the seminar (in-person and zoom). In case you are interested, please contact Dr. Sathaye
Talks:
- 8.5.2023 Amandine Escalier (WWU Münster) Orbit equivalence and graph products
- 15.5.2023 Francesco Russo (University of Cape Town) Topological entropy of locally compact groups
- 12.6.2023 in room SR 1B - Claudia Yun (MPI - Leipzig) Amalgamating groups via linear programming
- 19.6.2023 Lara Beßmann (WWU Münster) Universal groups and automatic continuity
- 26.6.2023 Camille Horbez (CNRS - Université Paris-Saclay) Measure equivalence rigidity among the Higman groups
- 3.7.2023 Gabriel Pallier (KIT) Large-scale geometry of rank one solvable Lie groups
- 10.7.2023 Markus Stroppel (Stuttgart) An incidence-geometric approach to an isomorphism of classical groups
- 17.7.2023 in room SR 1D Ludovic Mistiaen (Münster/Paris) A structural result on central groups
Abstract: We will present an ongoing project with Camille Horbez about the behaviour of graph products under (quantitative) orbit equivalence.
Abstract: The abstract can be found here.
Abstract: A compact group A is called an amalgamation basis if, for every way of embedding A into compact groups B and C, there exist a compact group D and embeddings B → D and C → D that agree on the image of A. Bergman in a 1987 paper studied the question of which groups can be amalgamation bases. A fundamental question that is still open is whether the circle group S1 is an amalgamation basis in the category of compact Lie groups. Further reduction shows that it suffices to take B and C to be the special unitary groups. In our work, we focus on the case when B and C are the special unitary group in dimension three. We reformulate the amalgamation question into an algebraic question of constructing specific Schur-positive symmetric polynomials and use integer linear programming to compute the amalgamation. We conjecture that S1 is an amalgamation basis based on our data. This is joint work with Michael Joswig, Mario Kummer, and Andreas Thom.
Abstract: Universal groups are subgroups of automorphism groups of right-angled buildings. Furthermore, they are totally disconnected locally compact topological groups and have interesting properties. I will present a finite presentation for discrete universal groups and results concerning the uniqueness of universal groups. Further, we will discuss the question of automatic continuity for universal groups. In particular, I will present conditions ensuring the continuity of abstract group homomorphisms into universal groups.
Abstract: In 1951, Higman introduced the first examples of infinite finitely presented groups without any nontrivial finite quotient. They have a simple presentation, with k ≥ 4 generators, where two consecutive generators (considered cyclically) generate a Baumslag-Solitar subgroup BS(1,2). Higman groups have received a lot of attention and remain mysterious in many ways. We study them from the viewpoint of measured group theory, and prove a superrigidity theorem in measure equivalence (a notion introduced by Gromov as a measurable analogue of quasi-isometry), as soon as the number k of generators is at least 5. I will explain the context and motivations, some consequences, and the main ideas from the proof. This is joint work with Jingyin Huang.
Abstract: Quasi-isometric rigidity is often, though not always, obtained by proving that a given group has few self quasi-isometries. In general, the solvable Lie groups (and their lattices) have more quasi-isometries than their semisimple counterparts. Nevertheless, reformulating works by other authors we can show that for a class of "rank one" solvable Lie groups, including the three-dimensional SOL, the quasi-isometries are often (and conjecturally, almost always) rough isometries, that is, they preserve any left-invariant Riemannian metric on the group up to a bounded error. This property is sometimes enough to deduce quasi-isometric rigidity, after some additional work. This is based on joint work with E. Le Donne and X. Xie; I will also review parts of recent works of T. Ferragut and of T. Dymarz, D. Fisher and X. Xie.
Abstract: In the classification of simple Lie algebras over the reals, several isomorphisms occur between algebras of small rank in different families. In particular, the real forms of the algebras of rank 3 and type A or D provide such examples. On the level of corresponding linear groups, some of these isomorphisms show up as homomorphisms from special unitary groups (of hermitian forms of Witt index 0,2,1 on a complex vector space of dimension 3, respectively) either onto connected components of orthogonal groups (of Witt index 0,2 on a real vector space of dimension 6) or the unitary group of a suitable hermitian form on a quaternionic vector space of dimension 3. In the latter case, the form is hermitian with respect to a non-standard involution on the quaternions. We construct the pertinent homomorphism via an isomorphism of incidence structures (namely, suitable polar unitals in complex projective space and the quaternion plane, respectively). This approach works well for all suitable ground fields (such that quaternion fields exist) as long as the characteristic is different from 2.
Abstract: A quick reasoning in abstract groups shows that if G/Z(G) is finite, then so is the derived subgroup G'. In this talk, we show that a similar result is true in the context of topological groups, replacing "finite" by "compact". This actually serves as an excuse to state and use a powerful structural result on central groups.