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Nonrational Genus Zero Function Fields and the Bruhat-Tits Tree

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Let K be a function field with constant field k, and let infinity be a fixed place of K. Let G be the Dedekind domain consisting of all those elements of K which are integral outside infinity. The group G = GL(2)(G) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat-Tits tree, J. Classical Bass-Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) G\J. The shape of this quotient graph (for any G) is described in a fundamental result of Serre. However, there are very few known examples for which a detailed description of G\J is known. (One such is the rational case, G = k[t], i.e., when K has genus zero, and infinity has degree one.) In this article, we give a precise description of G\J for the case where the genus of K is zero, K has no places of degree one, and infinity has degree two. Among the known examples a new feature here is the appearance of vertex stabilizer subgroups (of G) which are of quaternionic type.

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