Representations of currents taking values in $PGL(2,Q_q)$

Abstract

Let $G=PGL(2,Q_q)$. In this paper we shall investigate the group of measurable currents taking values in $G$. The key observation is that $G$ is acting by automorphisms on a homogeneous tree, which will play the role of the upper half plane in the case of $PSL(2,R)$.