Second Cohomology of the modular group

Question

What is known about $H^2(SL(2,\mathbb{Z}),\mathbb{Z})$? Anyone knows some references about that?

Answer 1

Yes, this is known, see the first lines here: "Our goal is to compute its cohomology groups with trivial coefficients, i.e. $H^q (SL_N(\mathbb{Z}),\mathbb{Z})$. The case $N = 2$ is well-known and follows from the fact that $SL_2(\mathbb{Z})$ is the amalgamated product of two finite cyclic groups ([21], [6], II.7, Ex.3, p.52). Here $[21]$ is Serre's book Trees.

I found the result also in Richard Hain's paper here, Proposition $3.13$, with proof (page $25-26$). $$ H^r(SL_2(\mathbb{Z}),\mathbb{Z})\cong \begin{cases} \mathbb{Z}/12, \text{ if $r$ is even} \\ 0, \text{otherwise} \end{cases} $$