Quasi-inner automorphisms of Drinfeld modular groups
A. W. Mason
University of Glasgow, Glasgow, UKAndreas Schweizer
Kongju National University, Gongju, South Korea
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Abstract
Let be the set of elements in an algebraic function field over which are integral outside a fixed place . Let be a Drinfeld modular group. The normalizer of in , where is the quotient field of , gives rise to automorphisms of , which we refer to as quasi-inner. Modulo the inner automorphisms of , they form a group which is isomorphic to , the -torsion in the ideal class group . The group acts on all kinds of objects associated with . For example, it acts freely on the cusps and elliptic points of . If is the associated Bruhat–Tits tree, the elements of induce non-trivial automorphisms of the quotient graph , generalizing an earlier result of Serre. It is known that the ends of are in one-to-one correspondence with the cusps of . Consequently, acts freely on the ends. In addition, acts transitively on those ends which are in one-to-one correspondence with the vertices of whose stabilizers are isomorphic to .
Cite this article
A. W. Mason, Andreas Schweizer, Quasi-inner automorphisms of Drinfeld modular groups. Groups Geom. Dyn. 18 (2024), no. 2, pp. 571–602
DOI 10.4171/GGD/765