Ping-pong configurations and circular orders on free groups

Abstract

We discuss actions of free groups on the circle with “ping-pong” dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains orMarkov partitions. Using this, we show that the free group $F_n$ admits an isolated circular order if and only if $n$ is even, in stark contrast with the case for linear orders. This answers a question from [21]. Inspired by work in [2], we also exhibit examples of “exotic” isolated points in the space of all circular orders on $F_2$. Analogous results are obtained for linear orders on the groups $F_n\times \mathbb{Z}$.