Dynamics on Thurston’s sphere of projective measured foliations.

*(English)*Zbl 0681.57002The paper is devoted to some dynamical properties of actions of subgroups of the mapping class group of a surface on Thurston’s sphere of projective classes of measured foliations. Let S be a compact surface of negative Euler characteristic. Let \(\Gamma\) be the mapping class group of S, MF be the space of (classes of) measured foliations on S, and PMF be the space of projective classes of measured foliations on S. We will denote by p: MF\(\to PMF\) the natural projection and by i(\(\cdot,\cdot)\) the intersection number of measured foliation. The main result of the paper can be stated as follows. Theorem. Let \(\Sigma\) be a subgroup of \(\Gamma\) containing two pseudo-Anosov elements f, g with no common power (i.e. \(f^ m\neq g^ n\) for all \(m,n\neq 0)\). Let \(\Lambda_ 0(\Sigma)\) be the set of fixed points in PMF of pseudo-Anosov elements in \(\Sigma\). Let \(\Lambda(\Sigma)\) be the closure of \(\Lambda_ 0(\Sigma)\), and \(Z\Lambda (\Sigma)=\{p(F):\) \(i(F,G)=0\) for some G with p(G)\(\in \Lambda (\Sigma)\}\). Then \(\Lambda(\Sigma)\) is the unique nonempty \(\Sigma\)- invariant minimal closed subset of PMF, \(\Sigma\) acts properly discontinuously on PMF\(\setminus Z\Lambda(\Sigma)\), and the measure of \(Z\Lambda(\Sigma)\setminus \Lambda(\Sigma)\) is zero. The paper contains also some natural versions of this theorem for other kinds of subgroups and a general discussion of related matters.

Reviewer’s remark. For the lemma 2.8 of this paper (it plays only an auxiliary role in the paper) the reader is referred only to an unpublished preprint of the first author. In fact, this result, asserting that every infinite irreducible subgroup contain a pseudo-Anosov element, was announced and discussed by the reviewer in his papers “Algebraic properties of mapping class group of surfaces” [Banach Cent. Publ. 18, 15-35 (1986; Zbl 0635.57004)] and “Subgroups of Teichmüller modular groups and their Frattini subgroups” [Funkts. Anal. Prilozh. 21, No.2, 76-77 (1987; Zbl 0629.57006)].

Reviewer’s remark. For the lemma 2.8 of this paper (it plays only an auxiliary role in the paper) the reader is referred only to an unpublished preprint of the first author. In fact, this result, asserting that every infinite irreducible subgroup contain a pseudo-Anosov element, was announced and discussed by the reviewer in his papers “Algebraic properties of mapping class group of surfaces” [Banach Cent. Publ. 18, 15-35 (1986; Zbl 0635.57004)] and “Subgroups of Teichmüller modular groups and their Frattini subgroups” [Funkts. Anal. Prilozh. 21, No.2, 76-77 (1987; Zbl 0629.57006)].

Reviewer: N.V.Ivanov

##### MSC:

57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |

57R50 | Differential topological aspects of diffeomorphisms |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |