Ping-Pong Configurations and Circular Orders on Free Groups

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 or Markov partitions. Using this, we show that the free group Fn 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 F2. Analogous results are obtained for linear orders on the groups Fn × ℤ. © European Mathematical Society