The stable homology of congruence subgroups

Calegari, Frank¹

¹ Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA

Geometry & topology, Tome 19 (2015) no. 6, pp. 3149-3191.

Résumé

We relate the completed cohomology groups of ${SL}_N\left({\mathcal{O}}_F\right)$, where ${\mathcal{O}}_F$ is the ring of integers of a number field, to $K$–theory and Galois cohomology. Various consequences include showing that Borel’s stable classes become infinitely $p$–divisible up the $p$–congruence tower if and only if a certain $p$–adic zeta value is nonzero. We use our results to compute $H_2\left({\Gamma }_N\left(p\right),{\mathbb{F}}_p\right)$ (for sufficiently large $N$), where ${\Gamma }_N\left(p\right)$ is the full level-$p$ congruence subgroup of ${SL}_N\left(\mathbb{Z}\right)$.

Détail

DOI : 10.2140/gt.2015.19.3149

Classification : 11F75, 19F99, 11F80
Keywords: arithmetic groups, stable homology, completed homology, $K$–theory

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