On the automorphisms of the Drinfeld modular groups

Abstract

Let $A$ be the ring of elements in an algebraic function field $K$ over $\mathbb{F}_q$ which are integral outside a fixed place $\infty$. In contrast to the classical modular group $SL_2(\mathbb{Z})$ and the Bianchi groups, the {\it Drinfeld modular group} $G=GL_2(A)$ is not finitely generated and its automorphism group $\mathrm{Aut}(G)$ is uncountable. Except for the simplest case $A=\mathbb{F}_q[t]$ not much is known about the generators of $\mathrm{Aut}(G)$ or even its structure. We find a set of generators of $\mathrm{Aut}(G)$ for a new case. \par On the way, we show that {\it every} automorphism of $G$ acts on both, the {\it cusps} and the {\it elliptic points} of $G$. Generalizing a result of Reiner for $A=\mathbb{F}_q[t]$ we describe for each cusp an uncountable subgroup of $\mathrm{Aut}(G)$ whose action on $G$ is essentially defined on the stabilizer of that cusp. In the case where $\delta$ (the degree of $\infty$) is $1$, the elliptic points are related to the isolated vertices of the quotient graph $G\setminus\mathcal{T}$ of the Bruhat-Tits tree. We construct an infinite group of automorphisms of $G$ which fully permutes the isolated vertices with cyclic stabilizer.