Quasi-inner automorphisms of Drinfeld modular groups.

Let A be the set of elements in an algebraic function field K over F_q which are integral outside a fixed place ∞. Let G = GL₂(A) be a Drinfeld modular group. The normalizer of G in GL₂(K), where K is the quotient field of A, gives rise to automorphisms of G, which we refer to as quasi-inner. Modulo the inner automorphisms of G, they form a group Quinn(G) which is isomorphic to Cl(A)₂, the 2-torsion in the ideal class group Cl(A). The group Quinn(G) acts on all kinds of objects associated with G. For example, it acts freely on the cusps and elliptic points of G. If T is the associated Bruhat--Tits tree, the elements of Quinn(G) induce non-trivial automorphisms of the quotient graph G\T, generalizing an earlier result of Serre. It is known that the ends of G\T are in one-to-one correspondence with the cusps of G. Consequently, Quinn(G) acts freely on the ends. In addition, Quinn.G/ acts transitively on those ends which are in one-to-one correspondence with the vertices of G\T whose stabilizers are isomorphic to GL₂(F_q). [ABSTRACT FROM AUTHOR]