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Let K be a function field of genus g with a finite constant field Fq . Choose a place ∞ of K of degree δ and let C be the arithmetic Dedekind domain consisting of all elements of K that are integral outside ∞. An explicit formula is given (in terms of q, g and δ) for the minimum index of a non-congruence subgroup in SL 2 (C). It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL 2 (C). The minimum index of a normal non-congruence subgroup is also determined.

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Abstract. Let C be a smooth projective curve over a field k. For each closed point Q of C let C = C(C, Q, k) be the coordinate ring of the affine curve obtained by removing Q from C. Serre has proved that GL 2 (C) is isomorphic to the fundamental group, π 1 (G, T ), of a graph of groups (G, T ), where T is a tree with at most one non-terminal vertex. Moreover the subgroups of GL 2 (C) attached to the terminal vertices of T are in one-one correspondence with the elements of Cl(C), the ideal class group of C. This extends an earlier result of Nagao for the simplest caseSerre's proof is based on applying the theory of groups acting on trees to the quotient graph X = GL 2 (C)\X, where X is the associated Bruhat-Tits building. To determine X he makes extensive use of the theory of vector bundles (of rank 2) over C. In this paper we determine X using a more elementary approach which involves substantially less algebraic geometry.The subgroups attached to the edges of T are determined (in part) by a set of positive integers S, say. In this paper we prove that S is bounded, even when Cl(C) is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of GL 2 (C), involving unipotent and elementary matrices.

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