A “Tits-alternative” for subgroups of surface mapping class groups.

*(English)*Zbl 0579.57006It has been observed that surface mapping class groups share various properties in common with the class of linear groups. In this paper the known list of such properties is extended to the “Tits-alternative”, a property of linear groups established by J. Tits.

Tits showed that every subgroup of \(\text{GL}(n,k)\), \(k\) a field of characteristic zero, either contains a solvable subgroup of finite index or a nonabelian free group. In the present paper it is proved that every subgroup of the mapping class group \(\Gamma (F)\) of a closed orientable surface \(F\) contains an abelian subgroup of finite index or a nonabelian free group (by a result of Birman, Lubotsky and the author every solvable subgroup of \(\Gamma (F)\) is virtually abelian). Moreover, for two generator subgroups of \(\Gamma (F)\) the following is established: given \(\sigma\),\(\tau\in \Gamma (F)\), some powers \(\sigma^ m\), \(\tau^ n\) either generate an abelian group or a nonabelian free group of rank two.

In order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston’s projective lamination spaces. This theory generalizes results known for pseudo-Anosov mapping classes.

The paper is carefully written, giving also a nice introduction to the background material on geodesic laminations and train tracks on surfaces.

Tits showed that every subgroup of \(\text{GL}(n,k)\), \(k\) a field of characteristic zero, either contains a solvable subgroup of finite index or a nonabelian free group. In the present paper it is proved that every subgroup of the mapping class group \(\Gamma (F)\) of a closed orientable surface \(F\) contains an abelian subgroup of finite index or a nonabelian free group (by a result of Birman, Lubotsky and the author every solvable subgroup of \(\Gamma (F)\) is virtually abelian). Moreover, for two generator subgroups of \(\Gamma (F)\) the following is established: given \(\sigma\),\(\tau\in \Gamma (F)\), some powers \(\sigma^ m\), \(\tau^ n\) either generate an abelian group or a nonabelian free group of rank two.

In order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston’s projective lamination spaces. This theory generalizes results known for pseudo-Anosov mapping classes.

The paper is carefully written, giving also a nice introduction to the background material on geodesic laminations and train tracks on surfaces.

Reviewer: B. Zimmermann

##### MSC:

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

57R50 | Differential topological aspects of diffeomorphisms |

20F38 | Other groups related to topology or analysis |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |