Geometry and Topology Seminar - Benjamin Brück - High-dimensional cohomology of special linear and symplectic groups

Description

Speaker: Benjamin Brück (ETH Zürich)

Title: High-dimensional cohomology of special linear and symplectic groups

Abstract: Computing the cohomology of arithmetic groups is a fundamental and often difficult problem at the intersection of topology, group theory and number theory. In this talk, I will explain how one can use duality phenomena to compute the rational cohomology of the arithmetic groups $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{Sp}_{2n}(\mathbb{Z})$ in "high" dimensions, i.e. close to their virtual cohomological dimension. Specifically, I will talk about joint work with Miller-Patzt-Sroka-Wilson in which we show that $H^{{n \choose 2} -2}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 4$. This was previously unknown, but confirms a conjecture of Church-Farb-Putman. In ongoing work with Patzt-Sroka, we are also trying to adapt these techniques to the group $\operatorname{Sp}_{2n}(\mathbb{Z})$.

Location: In-person

Room: Hamilton Hall 302
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