Peter PATZT - High dimensional cohomology of SL_n(Z) and its principal congruence subgroups 4 - Archive ouverte HAL Accéder directement au contenu
Vidéo Année : 2022

Peter PATZT - High dimensional cohomology of SL_n(Z) and its principal congruence subgroups 4

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Fanny Bastien

Résumé

Group cohomology of arithmetic groups is ubiquitous in the study of arithmetic K-theory and algebraic number theory. Rationally, SL_n(Z) and its finite index subgroups don't have cohomology above dimension n choose 2. Using Borel-Serre duality, one has access to the high dimensions. Church, Farb, and Putman conjectured that the high dimensional cohomology of SL_n(Z) with trivial rational coefficients vanishes. In this lecture series, we will give an introduction to these notions, prove the aforementioned conjecture in codimensions 0 and 1. We will also study the top cohomology of principal congruence subgroups. In the final lecture, we summarize some further directions and open problems in the field.

Dates et versions

hal-04662992 , version 1 (26-07-2024)

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  • HAL Id : hal-04662992 , version 1

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Peter Patzt, Fanny Bastien. Peter PATZT - High dimensional cohomology of SL_n(Z) and its principal congruence subgroups 4: Summer School 2022 - Cohomology, Geometry and Explicit Number Theory. 2022. ⟨hal-04662992⟩
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