Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, I: Construction

2019/06/09 by Juan Alonso, Sébastien Alvarez, Alonso, Juan +7
Mathematics · #20E06 #20E08 (Primary) #37C85 #57S05 (Secondary) #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1906.03578

openalex publication_date 2019/06/09 · openalex created_date 2019/06/14 · openalex updated_date 2026/07/28

Abstract

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group Diff^ω₊(\mathbb S¹) of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of Diff^ω₊(\mathbb S¹) acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403--453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work.

Citations

• Global rigidity of conjugations for locally non-discrete subgroups of Diff^w(S¹)
• Rigidity and flexibility of group actions on the circle
• Isolated circular orders of PSL(2,Z)

Related

• Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications
• Groups of fast homeomorphisms of the interval and the ping-pong argument
• Ping-pong configurations and circular orders on free groups
• Groups of Circle Diffeomorphisms
• Ping-pong dynamics of hyperbolic-like actions with non-simple points