Ping-pong configurations and circular orders on free groups

TL;DR

This paper explores ping-pong dynamics for free group actions on the circle, revealing that $F_n$ admits an isolated circular order if and only if n is even, and constructs exotic isolated points in the space of circular orders.

Contribution

It establishes a new criterion for the existence of isolated circular orders on free groups based on parity of n and introduces exotic isolated points in the space of circular orders.

Findings

Free groups $F_n$ have isolated circular orders if and only if n is even.

Constructs examples of exotic isolated points in the space of circular orders on $F_2$.

Provides analogous results for linear orders on $F_n imes bZ$.

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 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 (Mann, Rivas, 2016). Inspired by work of Alvarez, Barrientos, Filimonov, Kleptsyn, Malicet, Menino and Triestino, 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 $F_n \times \mathbb{Z}$.