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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 Raquel Murat.
Talks:
- 22.04.2025 Marjory Mwanza (Münster) On the isomorphism problem of Cayley graphs of graph products
- 29.04.2025 Emma Brink (Bonn) Condensed Group Cohomology
- 13.05.2025 Nora Hülskamp (Göttingen) Right-angled Artin groups and their outer automorphism groups
- 03.06.2025 Sruthy Joseph (Münster) Condensed abelian groups and duality
- 17.06.2025 Bianca Firmbach (Münster) Parabolic subgroups of large-type Artin groups
- 24.06.2025 Linus Kramer (Münster) Adjoint functors - why should I care?
Abstract: We investigate the isomorphism problem for Cayley graphs of graph products. We show that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs. Additionally, we identify conditions under which the Cayley graphs of two graph products are isomorphic, even when the underlying groups are not. This leads to interesting examples of non-isomorphic finitely generated groups with isomorphic Cayley graphs.
Abstract: Condensed mathematics provides a convenient framework for studying algebraic objects that carry a topology, and in particular enables the construction of a derived fixed point functor on continuous GG-modules for a topological group GG. In this talk, I will compare condensed group cohomology of a topological group with its continuous group cohomology (defined via continuous cochains), as well as with the (condensed/singular/sheaf) cohomology of its classifying space. For locally profinite groups and solid (e.g., locally profinite) continuous GG-modules, continuous group cohomology is isomorphic to condensed group cohomology. The same holds for locally compact, paracompact topological groups with finite-dimensional vector spaces as coefficients. In general, however, condensed group cohomology is a more refined invariant. I will explain that nonetheless, continuous group cohomology with solid coefficients can be described as Ext groups in the condensed setup for a broad class of groups.
Abstract: Automorphism groups of right-angled Artin groups (RAAGs) lie at the intersection of geometric group theory and algebraic topology and have been studied a lot. In this talk, we focus on computing the virtual cohomological dimension (vcd) of outer automorphism groups of RAAGs. I will present an algorithm recently developed by Day and Wade that enables such computations. The algorithm breaks the outer automorphism group down using short exact sequences and can be visualized as a tree-like structure, reflecting the recursive decomposition process. It is quite intricate, and I will illustrate the method using concrete examples.
Abstract: Condensed mathematics was developed by Dustin Clausen and Peter Scholze to address difficulties in doing algebra on algebraic structures that carry a topology. For instance, we cannot do homological algebra in the category of topological abelian groups because it does not form an abelian category. This necessity has given rise to the category of condensed abelian groups, which forms an abelian category. In the first half of the talk, I am reformulating the definition of condensed sets in the language of Boolean rings using Stone duality. The second half focuses on discussing the properties of the category of condensed abelian groups. I also propose a candidate for the Pontryagin dual of a condensed abelian group and demonstrate why this choice of dual does not satisfy Pontryagin duality in the category of condensed abelian groups.
Abstract: Artin groups generalize braid groups and are closely related to Coxeter groups, yet their structure and geometry remain less well understood. A recent geometric approach by Cumplido, Martin, and Vaskou has provided new insights into the structure of parabolic subgroups of large-type Artin groups. The central tool in this study is the Artin complex—a simplicial complex on which the Artin group acts cocompactly and without inversions, with parabolic subgroups appearing as stabilizers of simplices. For large-type Artin groups on at least three generators, the associated Artin complex admits a systolic geometry, which provides a powerful framework for understanding parabolic subgroups. We establish that parabolic subgroups are closed under arbitrary intersections, form a lattice with respect to inclusion, and we describe their normalizers. As an application, we can solve the conjugacy stability problem for parabolic subgroups of large-type Artin groups and show that they are stable under taking roots.
Abstract: This will be an elementary talk about limits, colimits, universal arrows and adjoint functors. I will give some examples and try to motivate why it might be useful to recognize an adjoint functor.