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

TL;DR

This paper classifies the topological dynamics of certain discrete groups of real-analytic circle diffeomorphisms, extending ping-pong techniques and solving a longstanding conjecture in foliation theory.

Contribution

It introduces a new classification method for locally discrete, finitely generated groups of circle diffeomorphisms and extends ping-pong lemma to actions of graphs of groups.

Findings

Classification of topological dynamics of these groups

Extension of ping-pong lemma to graphs of groups

Resolution of Dippolito's conjecture in this setting

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 $\mathsf{Diff}^\omega_+(\mathbb S^1)$ of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of $\mathsf{Diff}^\omega_+(\mathbb S^1)$ 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.