Classifying Spaces of Matrix Lie Groups

Question

I’m currently studying the classifying spaces of some of the matrix Lie groups. I’ve come across a post here that describes the classifying spaces for $SO(n)$, $SU(n)$, $GL(n)$, and $Sp(n)$. I’ve also seen computations for these in other places.

I'm interested in the classifying space for $SL(n,\mathbb{Z})$. As discussed in the comments, this is the same as the Eilenberg-MacLane space $K(SL(n,\mathbb{Z}),1)$. There is a general construction of these spaces given in Hatcher's algebraic topology. However, I can't determine from this if $K(SL(n,\mathbb{Z}),1)$ is some common space. Is there a different method that determines $K(SL(n,\mathbb{Z}),1)$?

Answer 1

Here is an answer of sorts. The quotient $X=SL(n,{\mathbb R})/SO(n)$ is a symmetric space of noncompact type, hence, a contractible manifold. The group $\Gamma=SL(n,{\mathbb Z})$ acts on $X$ properly discontinuously but not freely (via the left multiplication of cosets): Stabilizers of points are finite. The group $\Gamma$ contains a torsion-free subgroup $\Gamma_0$ of finite index, the kernel of the natural projection $$ \Gamma\to SL(n,{\mathbb Z}/3). $$ (I think, this is due to I.Schur.) Hence, $\Gamma_0\backslash X$ is a classifying space for $\Gamma_0$. To get one for $\Gamma$ itself one can do the following. Take the classifying space for the finite group $SL(n,{\mathbb Z}/3)$, it is the quotient $SL(n,{\mathbb Z}/3)\backslash Y$, where $Y$ is a contractible CW complex. (This should be treated as a black box.) You get a free proper action of $\Gamma$ on $X\times Y$ where the group $\Gamma$ acts on $X$ as above and acts on $Y$ via the homomorphism $\Gamma\to SL(n,{\mathbb Z}/3)$. Then the quotient $$ \Gamma\backslash (X\times Y) $$ is a classifying space for $\Gamma$. What is it good for, I do not know.