Normal subgroups of $SL_2$ of a polynomial ring

Question

What is known about normal subgroups of $SL_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction modulo some ideal of $\mathbb{C}[X]$?

Answer 1

[EDIT] These groups have been studied for a long time from various viewpoints, so there is a long paper-trail. I'd emphasize however that working over the complex numbers is usually similar to working over an arbitrary infinite field. Finite fields on the other hand occur more often in arithmetic contexts.

Concerning normal subgroups, the problem looks essentially hopeless, just as in the closely parallel situation of $\mathrm{SL}_2(\mathbb{Z})$ (or its quotient the modular group) first studied over a century ago. There the congruence subgroup problem has a strongly negative answer, in a sense eventually made precise by Serre. In the modern study of the congruence subgroup problem (Bass-Milnor-Serre and beyond), the most manageable situation involves higher rank algebraic groups. But Serre did obtain almost definitive results for rings of integers (and closely related rings of arithmetic interest). His paper is now available online through JSTOR: Le probleme des groupes de congruence pour SL2. Ann. of Math. (2) 92 1970 489–527.

While the congruence subgroup problem is studied mainly in the arithmetic setting, the normal subgroup problem is complicated in a similar way because the matrix group (in your case over a polynomial ring in one variable) has a huge number of such subgroups. In the case of the modular group, these arise from the group-theoretic structure as an amalgamated free product of two small cyclic groups. The analogue in your question is worked out in the paper by Hirosi Nagao, *On $\mathrm{GL}(2, K[x])$, J. Inst. Polytech. Osaka City Univ. Ser. A 10 (1959) 117–121. In most of the literature, attention is focused instead on the somewhat better behaved classical (or Chevalley) groups of higher Lie rank than 1.

While the cumulative literature on matrix groups over rings of various types is enormous and spreads out into algebraic $K$-theory, I'll mention a few of the many people involved over the years: E. Abe, W. Klingenberg, A.W. Mason, A.A. Suslin, L.N. Vaserstein, N.A. Vavilov. Here is a randomly chosen paper: A.W. Mason, Anomalous normal subgroups of $\mathrm{SL}_2(K[x])$. Quart. J. Math. Oxford Ser. (2) 36 (1985), no. 143, 345–358. Vavilov has written many papers on general structure theory, in both Russian and English, including some long surveys.

P.S. Though I'm not a specialist on structure of linear groups over rings, all evidence I've seen confirms that determining the normal subgroups of groups $\mathrm{SL}_2(K[x])$ (or $\mathrm{GL}_2(K[x])$) when $K$ is a field is basically hopeless. Even though the ideal structure of the ring itself is reasonable (as for $\mathbb{Z}$), Nagao's old work already shows how far that is from locating all normal subgroups. Much of the literature is covered in the 1989 Springer book by Hahn-O'Meara The Classical Groups and $K$-Theory (Chapter 4). The papers by Costa-Keller get some mileage in special cases from the refined notion of "radix", but for the rings here every radix is actually an ideal unless $K$ has characteristic 2.

Answer 2

Most likely, the analog of the standard "sandwich classification" of normal subgroups of $SL_2(\mathbb{C}[X])$ involves the notion of a radix, which gives the more careful form of a congruence subgroup.

See the paper D. Costa, G. Keller, Normal subgroups of $SL(2,A)$, where this is done for $A$ a ring of stable rank 1 or a Dedekind ring of arithmetic type. $\mathbb{C}[X]$ is neither of them, but still of rank 2, so the ideas from the paper might be helpful.