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

It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g., [ B L M , H ] [BLM,\,H] ). In this paper, the known list of such properties is extended to the “Tits-Alternative”, a property of linear groups established by J. Tits [ T ] [T] . In fact, we establish that every subgroup of a surface mapping class group is either virtually abelian or contains a nonabelian free group. In addition, 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 [ T h 1 ] [Th1] . This theory generalizes results known for pseudo-Anosov mapping classes [ F L P ] [FLP] .