Why is SL(n,Z)[p] modulo the group normally generated by elementary matrices abelian?

Question

I'm trying to understand part of Bass-Milnor-Serre's paper on the congruence subgroup problem. I'm pretty sure that the following statement is proved in there, but I'm having trouble find (and/or understanding) the proof. Can someone help me?

Fix some $\ell \geq 2$. Let $\Gamma_n$ be $\text{SL}(n,\mathbb{Z})$, let $\Gamma_n(\ell)$ be the level $\ell$ principal congruence subgroup of $\text{SL}(n,\mathbb{Z})$, and let $E_n(\ell)$ be the normal closure in $\Gamma_n$ of the set of elementary matrices whose off-diagonal entry is $\ell$. The thing I want to understand then is that for $n \geq 3$, we have $[\Gamma_n,\Gamma_n(\ell)] \subset E_n(\ell)$. In particular, $\Gamma_n(\ell)/E_n(\ell)$ is an abelian group.

Answer 1

I think I can help locating the statement in the Bass-Milnor-Serre paper. First, if I understand your question correctly, the translation between your notation and the one of Bass-Milnor-Serre is $\Gamma_n(\ell)=\operatorname{SL}_n(\mathbb{Z},(\ell))$, $E_n(\ell)=E_n(\mathbb{Z},(\ell))$, and in particular $\Gamma_n(\ell)/E_n(\ell)=C_{(\ell)}(n)$. The commutator formula $E_n(\ell)=[\Gamma_n,\Gamma_n(\ell)]$ is then part a) of Theorem 4.1 (on p. 94). Then Corollary 4.3 provides an explicit computation of $C_{\mathfrak{q}}$ for arbitrary ideals in arithmetic Dedekind rings, which in the special case of $\mathbb{Z}$ implies $C_{(\ell)}(n)=\{1\}$, i.e., $E_n(\mathbb{Z},(\ell))=\operatorname{SL}_n(\mathbb{Z},(\ell))$.

I am not sure if I can help with understanding the proof without a more specific explanation of the problems you are having with it. An outline of the proof strategy can be found in the introduction:

• The first step is to find a set of generators. Philosophically, you are dealing with a relative $K_1$-group and there is a stabilization theorem which implies (for Dedekind rings, i.e. Krull dimension $1$) that everything is generated by stuff from $\operatorname{SL}_2$ and elementary matrices.

• The elementary matrices behave exactly like in the formula $E_n(R)=[GL_n(R),GL_n(R)]$ resp. in the proof that $K_1(R)$ is abelian.

• So you have to deal with the generators from $\operatorname{SL}_2$, which is done via the theory of Mennicke symbols. This then completely describes the relative $K_1$ group $\operatorname{SL}_n(A,\mathfrak{q})/E_n(A,\mathfrak{q})$. The result for $\mathbb{Z}$ (or more generally case b) of Corollary 4.3) is proved by showing (using the defining relations) that all Mennicke symbols have to be trivial - via explicit computations with Mennicke symbols and quite some number theory.

Finally, I have to point to a very powerful generalization of the Bass-Milnor-Serre result in A. Bak, R. Hazrat and N. Vavilov: Localization-completion strikes again: relative $K_1$ is nilpotent by abelian. JPAA 213 (2009), 1075-1085. As the title says, the paper contains a very general result on the structure of groups like $\Gamma_n(\ell)/E_n(\ell)$ for other groups than $\operatorname{SL}_n$ or $\operatorname{Sp}_{2n}$, other rings than just Dedekind rings etc. There is a huge amount of related literature on the subgroup structure of Chevalley groups over rings, see the bibliography of that paper...