Let v be a (discrete) valuation of the rational function field k(t), where k is a field, and let b(v) be the intersection of the valuation rings of all the valuations of k(t), other than that of v. It is well-known that either v = v(pi) the valuation determined by some irreducible polynomial pi in k[t], or v = v(infinity), the "valuation at infinity". In this paper we prove that GL(2)(b(v)) where v = v(pi) is the fundamental group of a certain tree of groups. The tree has finitely many vertices and its terminal vertices correspond with the elements of the ideal class group of b(v) This extends a previous result of Nagao for the special case v = v.. In this case 16, = k[t] and Nagao proves that GL(2)(k[t]) is an amalgamated product of a pair of groups. As a consequence we show that, when the degree of pi is at least 4, GL(2)(b(v)) has a free, non-cyclic quotient whose kernel contains (for example) all the unipotent matrices. This represents a two-dimensional anomaly.
