On the automorphisms of the Drinfeld modular groups

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 contrastto 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 itsautomorphism group $\mathrm{Aut}(G)$ is uncountable. Except for the simplestcase $A=\mathbb{F}_q[t]$ 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 foreach cusp an uncountable subgroup of $\mathrm{Aut}(G)$ whose action on $G$ isessentially defined on the stabilizer of that cusp. In the case where $\delta$(the degree of $\infty$) is $1$, the elliptic points are related to theisolated vertices of the quotient graph $G\setminus\mathcal{T}$ of theBruhat-Tits tree. We construct an infinite group of automorphisms of $G$ whichfully permutes the isolated vertices with cyclic stabilizer.