Cohomology of lattice subgroups

Question

I am trying to find a reference for lower cohomology groups $H^i(G, \mathbb{Z}),$ for $i=1, 2, 3$ for lattices in higher rank (for example, $SL(n, \mathbb{Z}), Sp(2n, \mathbb{Z}),$ and possibly congruence subgroups thereof. There is the Borel periodicity formula, but that's over $\mathbb{Q}\dots$

Answer 1

I don't think this is known in complete generality. Here are some results. Everything is easy in rank 1, so I'll restrict myself to $SL_n(\mathbb{Z})$ for $n \geq 3$ and $Sp_{2g}(\mathbb{Z})$ for $g \geq 2$. Denote the level $L$ congruence subgroups of $SL_n(\mathbb{Z})$ and $Sp_{2g}(\mathbb{Z})$ by $SL_n(\mathbb{Z},L)$ and $Sp_{2g}(\mathbb{Z},L)$.

First homology

Of course, both $SL_n(\mathbb{Z})$ and $Sp_{2g}(\mathbb{Z})$ are perfect, so $H_1$ of them are trivial. This can be found in many places -- a suitable textbook reference (that definitely includes the symplectic group, which many sources omit) is Hahn-O'Meara's book "The classical groups and K-theory".

For the congruence subgroups, it is proven in

R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976), no. 1, 15–53.

that $H_1(SL_n(\mathbb{Z},L)) \cong \mathfrak{sl}_n(\mathbb{Z}/L)$, the abelian group of $n \times n$ matrices over $\mathbb{Z}/L$ with trace $0$. The associated abelian quotient is easy to construct -- an arbitrary element of $SL_n(\mathbb{Z},L)$ is of the form $1+L M$ for some matrix $M$, and the associated homomorphism $\phi : SL_n(\mathbb{Z},L) \rightarrow \mathfrak{sl}_n(\mathbb{Z})$ is defined by $\phi(1+LM)=M$ modulo $L$. This is a homomorphism since $$(1+LM)(1+LN) = 1+L(M+N)+L^2 MN.$$ This was extended to prove that for $L$ odd, $H_1(Sp_{2g}(\mathbb{Z},L)) \cong \mathfrak{sp}_{2g}(\mathbb{Z}/L)$ independently in

B. Perron, Filtration de Johnson et groupe de Torelli modulo p, p premier, C. R. Math. Acad. Sci. Paris 346 (2008), no. 11-12, 667–670.

and

A. Putman, The abelianization of the level L mapping class group, arXiv:0803.0539

and

M. Sato, The abelianization of the level d mapping class group, J Topology (2010) 3 (4): 847-882.

For $L$ even, Sato in the last paper above proves that the abelianization of $Sp_{2g}(\mathbb{Z},L)$ is an extension of $\mathfrak{sp}_{2g}(\mathbb{Z}/L)$ by $H_1(\Sigma_g;\mathbb{Z}/2)$, where $\Sigma_g$ is a genus $g$ surface.

Second homology

Let me now turn to the second homology group. For the special linear group, it follows from Corollary 10.2 and the remark after Theorem 5.10 of Milnor's book on algebraic k-theory that $H_2(SL_n(\mathbb{Z})) \cong \mathbb{Z}/2$ for $n \geq 5$. For $n=3,4$, it is proven in

W. van der Kallen, The Schur multipliers of SL(3,Z) and SL(4,Z), Math. Ann. 212 (1974/75), 47-49.

that $H_2(SL_n(\mathbb{Z})) \cong \mathbb{Z}/2 \oplus \mathbb{Z}/2$.

It is also known that for $g$ at least $4$, we have $H_2(Sp_{2g}(\mathbb{Z})) \cong \mathbb{Z}$. This is an old result, but I don't know a great reference for it. Because of this, I gave a (somewhat lame) proof of it in my paper "The Picard group of the moduli space of curves with level structures". See the remark after the proof of Lemma 7.5 of that paper. For $g = 3$, it is proven in

M. Stein, The Schur multipliers of Sp6(Z), Spin8(Z), Spin7(Z), and F4(Z), Math Ann., Volume 215, Number 2, 165-172.

that $H_2(Sp_{2g}(\mathbb{Z})) \cong \mathbb{Z} \oplus \mathbb{Z}/2$.

I don't know any general results for the second homology groups of congruence subgroups. The only calculation I am aware of is in

R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976), no. 1, 15–53.

which calculates $H_2(SL_{3}(\mathbb{Z},3))$.

Of course, for $n$ large (I think that $n \geq 4$ should work) Borel's theorem tells you that $H^2(SL_n(\mathbb{Z},L))$ has rank $0$, so you can determine it integrally using the above calculation of $H_1$. A similar remark applies to the symplectic group. However, I'm pretty sure that $H_2$ is not known intergrally for congruence subgroups except in sporadic cases.

Third homology

The only results I am aware of for $H_3$ are the rational calculations that follow from Borel's work and Soule's complete calculation of the cohomology ring of $SL_3(\mathbb{Z})$ in

C. Soule, The cohomology of SL3(Z), Topology 17 (1978) 1–22