O seminarium

Terminy i tematyka spotkań

poniedziałek, 06-02-2023 - 16:15, WS
Computer proofs for Property (T), and SDP duality
Martin Nitsche
Kazhdan's Property (T) is a strong rigidity property for groups. In recent years a new method has been established for proving Property (T) with the computer. I will explain this approach from the perspective of the corresponding dual optimization problem, which has a geometric interpretation in terms of harmonic cocycles. This viewpoint can be used to simplify the computer calculation, making it feasible to prove Property (T) for $Aut(F_4)$.
poniedziałek, 30-01-2023 - 16:15, WS
Algebraic fibring and finite quotients
Sam Hughes (University of Oxford)
A number of remarkable recent results in profinite rigidity use a theorem of Friedl and Vidussi that connects fibring of 3-manifolds with non-vanishing of twisted Alexander polynomials. One such result, due to Liu, states that there are only finitely many finite volume hyperbolic 3-manifolds whose fundamental groups have the same set of finite quotients. In the first part of this talk, based on joint work with Dawid Kielak, we will look at relatives of the theorem of Friedl and Vidussi for LERF groups and it's connections to profinite rigidity. In the second part of the talk, based on ongoing joint work with Monika Kudlinska, we will discuss an analogue of Liu's result for irreducible hyperbolic free-by-cyclic groups.
poniedziałek, 23-01-2023 - 16:15, WS
The cohomology of classical arithmetic groups
Yuri Santos
Abstract: Cohomological results for arithmetic groups and related spaces, such as symmetric spaces and buildings, are of paramount importance in algebra, geometry, and number theory. In this setting, a famous result of Lee--Szczarba states that the cohomology of $SL(n,\mathbb{Z})$ with rational coefficients is zero in dimension $n(n-1)/2$. The first half of this talk will be an overview on cohomology computations for arithmetic groups, with focus on the work of Borel and Serre on associated symmetric spaces and duality. We will then focus on arithmetic subgroups of classical semisimple groups, such as symplectic and orthogonal groups, and shall see how to combine the Borel--Serre construction with a result of Tóth to obtain a generalization of the theorem of Lee--Szczarba: the rational cohomology of such arithmetic groups vanishes in their virtual cohomological dimension. This is based on joint work with B. Brück and R. Sroka
czwartek, 22-12-2022 - 17:00, WS
Perfect matchings in hyperfinite graphings
Marcin Sabok
The talk will focus on recent results on measurable perfect matchings in hyperfintie graphings. We will start by defining hyperfinite graphings and recall some motivations behind this definition, such as the Benjamini-Schramm limits and hyperfinite sequences of graphs. As the main result we will discuss the recent theoremt saying that every regular hyperfinite one-ended bipartite graphing admits a measurable perfect matching. We will see some applications of this results, answering several questions in the field. For instance we will characterize the existence of factor of iid perfect matchings in bipartite Cayley graphs, extending a result of Lyons and Nazarov. We will also answer a question of Bencs, Hruskova and Toth arising in the study of balanced orientations in graphings. Finally, we see how the results imply the measurable circle squaring. This is joint work with Matt Bowen and Gabor Kun.
czwartek, 22-12-2022 - 15:45, WS
Tits alternative for the 3-dimensional tame automorphism group
Piotr Przytycki
Let $k$ be a field of characteristic $0$. The tame automorphism group $\mathrm{Tame}(k^3)$ is the group generated by the affine maps of $k^3$ and the maps of form $(x,y,z)\rightarrow (x,y,z+P(x,y))$. We prove the strong Tits alternative for $\mathrm{Tame}(k^3)$, using its action on a $2$-dimensional CAT$(0)$ complex. This is joint work with Stéphane Lamy.
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