6533b863fe1ef96bd12c794c

RESEARCH PRODUCT

Ping-pong configurations and circular orders on free groups

Cristóbal RivasKathryn MannMichele TriestinoDominique Malicet

subject

[ MATH.MATH-GR ] Mathematics [math]/Group Theory [math.GR][ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS][MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]MSC2010: Primary 20F60 57M60. Secondary 20E05 37C85 37E05 37E10 57M60.Extension (predicate logic)Group Theory (math.GR)Dynamical Systems (math.DS)Space (mathematics)20F60 57M60[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]CombinatoricsFree groupsOne-dimensional dynamicsFree groupPing pongFOS: MathematicsDiscrete Mathematics and CombinatoricsOrder (group theory)Geometry and TopologyMathematics - Dynamical SystemsMathematics - Group TheoryMathematicsOrders on groups

description

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 $F_n \times \mathbb{Z}$.

yearjournalcountryeditionlanguage
2017-09-07
https://dx.doi.org/10.48550/arxiv.1709.02348