A Group of Automorphisms of the Rooted Dyadic Tree and Associated Gelfand Pairs

Abstract

In this paper we study the group $G$ of automorphisms of the rooted dyadic tree generated by translation by 1 and multiplication by an odd integer $q$, showing that $G$ admits a self-similar presentation and it is isomorphic to the Baumslag–Solitar group ${BS}_q$. Moreover, we show that the action of $G$ on each level of the tree gives rise to a Gelfand pair.