On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ)

Miller, Jeremy¹ ; Patzt, Peter¹ ; Putman, Andrew²

¹ Department of Mathematics, Purdue University, West Lafayette, IN, United States
² Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States

Geometry & topology, Tome 25 (2021) no. 2, pp. 999-1058.

Résumé

Let ${\Gamma }_n\left(p\right)$ be the level-$p$ principal congruence subgroup of ${SL}_n\left(\mathbb{Z}\right)$. Borel and Serre proved that the cohomology of ${\Gamma }_n\left(p\right)$ vanishes above degree $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$. We study the cohomology in this top degree $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$. Let ${\mathcal{𝒯}}_n\left(\mathbb{Q}\right)$ denote the Tits building of ${SL}_n\left(\mathbb{Q}\right)$. Lee and Szczarba conjectured that $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\Gamma }_n\left(p\right)\right)$ is isomorphic to ${\tilde{H}}_{n-2}\left({\mathcal{𝒯}}_n\left(\mathbb{Q}\right)∕{\Gamma }_n\left(p\right)\right)$ and proved that this holds for $p=3$. We partially prove and partially disprove this conjecture by showing that a natural map $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\Gamma }_n\left(p\right)\right)\to {\tilde{H}}_{n-2}\left({\mathcal{𝒯}}_n\left(\mathbb{Q}\right)∕{\Gamma }_n\left(p\right)\right)$ is always surjective, but is only injective for $p\le 5$. In particular, we completely calculate $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\Gamma }_n\left(5\right)\right)$ and improve known lower bounds for the ranks of $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)}\left({\Gamma }_n\left(p\right)\right)$ for $p\ge 5$.

Détail

DOI : 10.2140/gt.2021.25.999

Classification : 11F75
Keywords: congruence subgroups, Steinberg module, cohomology of arithmetic groups

Bibliographie

[1] A Adem, Buildings, group extensions and the cohomology of congruence subgroups, from: "Homotopy theory via algebraic geometry and group representations" (editors M Mahowald, S Priddy), Contemp. Math. 220, Amer. Math. Soc. (1998) 1 | DOI

[2] A Ash, Cohomology of subgroups of finite index of SL(3,Z) and SL(4,Z), Bull. Amer. Math. Soc. 83 (1977) 367 | DOI

[3] A Ash, P E Gunnells, M Mcconnell, Cohomology of congruence subgroups of SL4(Z), J. Number Theory 94 (2002) 181 | DOI

[4] A Ash, P E Gunnells, M Mcconnell, Cohomology of congruence subgroups of SL(4, Z), II, J. Number Theory 128 (2008) 2263 | DOI

[5] A Ash, P E Gunnells, M Mcconnell, Cohomology of congruence subgroups of SL4(Z), III, Math. Comp. 79 (2010) 1811 | DOI

[6] H Bass, M Lazard, J P Serre, Sous-groupes d’indice fini dans SL(n,Z), Bull. Amer. Math. Soc. 70 (1964) 385 | DOI

[7] M Bestvina, PL Morse theory, Math. Commun. 13 (2008) 149

[8] A Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. 7 (1974) 235 | DOI

[9] A Borel, J P Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436 | DOI

[10] K S Brown, Buildings, Springer (1989) | DOI

[11] V A Bykovskii, Generating elements of the annihilating ideal for modular symbols, Funktsional. Anal. i Prilozhen. 37 (2003) 27 | DOI

[12] R Charney, A generalization of a theorem of Vogtmann, J. Pure Appl. Algebra 44 (1987) 107 | DOI

[13] T Church, A Putman, The codimension-one cohomology of SLnZ, Geom. Topol. 21 (2017) 999 | DOI

[14] R C Gunning, Lectures on modular forms, 48, Princeton Univ. Press (1962)

[15] A Hatcher, N Wahl, Stabilization for mapping class groups of 3–manifolds, Duke Math. J. 155 (2010) 205 | DOI

[16] W Van Der Kallen, Homology stability for linear groups, Invent. Math. 60 (1980) 269 | DOI

[17] R Lee, J Schwermer, Cohomology of arithmetic subgroups of SL3 at infinity, J. Reine Angew. Math. 330 (1982) 100 | DOI

[18] R Lee, R H Szczarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976) 15 | DOI

[19] J L Mennicke, Finite factor groups of the unimodular group, Ann. of Math. 81 (1965) 31 | DOI

[20] J Miller, P Patzt, J C H Wilson, D Yasaki, Non-integrality of some Steinberg modules, J. Topol. 13 (2020) 441 | DOI

[21] A Paraschivescu, On a generalization of the double coset formula, Duke Math. J. 89 (1997) 1 | DOI

[22] D Quillen, Homotopy properties of the poset of nontrivial p–subgroups of a group, Adv. Math. 28 (1978) 101 | DOI

[23] L Solomon, The Steinberg character of a finite group with BN–pair, from: "Theory of finite groups" (editors R Brauer, C h Sah), Benjamin (1969) 213