Ping-pong configurations and circular orders on free groups (Q2286348)

Summary: 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 \(F_n\) 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 [\textit{K. Mann} and \textit{C. Rivas}, Ann. Inst. Fourier 68, No. 4, 1399--1445 (2018; Zbl 07002300)]. Inspired by work in [\textit{S. Alvarez} et al., ``Generalized ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms. II: Applications'', Preprint, \url{arXiv:2104.03348}], 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 \(F_n \times \mathbb{Z}\).