Isolated circular orders of PSL(2,$\mathbb{Z}$)

Abstract

We give a bijection between the isolated circular orders of the group $G=\mathrm{PSL}(2,\mathbb{Z})\approx (\mathbb{Z}/2\mathbb{Z})\ast (\mathbb{Z}/3\mathbb{Z})$ and the equivalence classes of Markov systems associated to $G$. As applications, we present examples of isolated circular orders of the group $G$.