Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403–453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type. MSC2020: Primary 37C85, 20E06. Secondary 20E08, 37B05, 37E10.

[1]  M. R. Herman.  Sur la Conjugaison Différentiable des Difféomorphismes du Cercle a des Rotations , 1979 .

[2]  J. Rebelo.  Ergodicity and rigidity for certain subgroups of Diffω(S1) , 1999 .

[3]  A. Navas.  GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS , 2017, Proceedings of the International Congress of Mathematicians (ICM 2018).

[4]  Michele Triestino,et al.  Ping-pong configurations and circular orders on free groups , 2017, Groups, Geometry, and Dynamics.

[5]  T. Koberda,et al.  Flexibility of Group Actions on the Circle , 2016, Lecture Notes in Mathematics.

[6]  David Gabai.  Convergence groups are Fuchsian groups , 1991 .

[7]  Paul R. Dippolito.  Codimension one foliations of closed manifolds , 1978 .

[8]  R. Bieri,et al.  On Groups of PL-homeomorphisms of the Real Line , 2014, 1411.2868.

[9]  É. Ghys,et al.  Sur un groupe remarquable de difféomorphismes du cercle , 1987 .

[10]  É. Ghys,et al.  A Singular Mathematical Promenade , 2016 .

[11]  Michele Triestino,et al.  PING-PONG PARTITIONS AND LOCALLY DISCRETE GROUPS OF REAL-ANALYTIC CIRCLE DIFFEOMORPHISMS, I: CONSTRUCTION , 2019, 1906.03578.

[12]  A. Navas,et al.  On the ergodic theory of free group actions by real-analytic circle diffeomorphisms , 2013, 1312.4133.

[13]  Igor Nikolaev.  Arithmetic of hyperbolic 3-manifolds , 2002 .

[14]  P. Tukia.  Homeomorphic conjugates of Fuchsian groups. , 1988 .

[15]  Locally discrete expanding groups of analytic diffeomorphisms of the circle , 2018, Journal of Topology.

[16]  É. Ghys.  Classe d'euler et minimal exceptionnel , 1987 .

[17]  J. Rebelo,et al.  Global rigidity of conjugations for locally non-discrete subgroups of Diff^w(S^1) , 2015, 1507.03855.

[18]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[19]  R. Langevin.  A List of Questions about Foliations , 2000 .

[20]  Michele Triestino,et al.  Small C1 actions of semidirect products on compact manifolds , 2019, Algebraic & Geometric Topology.

[21]  V. Kleptsyn,et al.  One-end finitely presented groups acting on the circle , 2014 .

[22]  Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function , 2008, 0811.0192.

[23]  A. Navas,et al.  Groups with infinitely many ends acting analytically on the circle , 2015, Journal of Topology.

[24]  A. Navas.  ON UNIFORMLY QUASISYMMETRIC GROUPS OF CIRCLE DIFFEOMORPHISMS , 2006 .

[25]  D. Jungreis,et al.  Convergence groups and seifert fibered 3-manifolds , 1994 .