Abstract
Let ν be a (discrete) valuation of the rational function field k(t), where k is a field, and let 𝒞ν be the intersection of the valuation rings of all the valuations of k(t), other than that of ν. It is well-known that either ν = νπ, the valuation determined by some irreducible polynomial π in k[t], or ν = ν∞, the “valuation at infinity”. In this paper we prove that GL 2(𝒞ν), where ν = νπ, 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 𝒞ν. This extends a previous result of Nagao for the special case ν = ν∞. In this case 𝒞ν = 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 π is at least 4, GL 2(𝒞ν) has a free, non-cyclic quotient whose kernel contains (for example) all the unipotent matrices. This represents a two-dimensional anomaly.
Mathematics Subject Classification 2000: