Groups, Geometry, and Dynamics

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Volume 13, Issue 4, 2019, pp. 1195–1218
DOI: 10.4171/GGD/519

Published online: 2019-09-27

Ping-pong configurations and circular orders on free groups

Dominique Malicet[1], Kathryn Mann[2], Cristóbal Rivas[3] and Michele Triestino[4]

(1) Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France
(2) Cornell University, Ithaca, USA
(3) Universidad de Santiago de Chile, Chile
(4) Université de Bourgogne Franche-Comté, Dijon, France

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 $F_n \times \mathbb Z$.

Keywords: Free groups, left-invariant order, actions on one-dimensional manifolds

Malicet Dominique, Mann Kathryn, Rivas Cristóbal, Triestino Michele: Ping-pong configurations and circular orders on free groups. Groups Geom. Dyn. 13 (2019), 1195-1218. doi: 10.4171/GGD/519