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

McCarthy, John

doi:10.1090/S0002-9947-1985-0800253-8

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

Trans. Amer. Math. Soc. 291 (1985), 583-612

DOI: https://doi.org/10.1090/S0002-9947-1985-0800253-8

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Abstract:

It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g., $[\mathbf {BLM}, \mathbf {H}]$). In this paper, the known list of such properties is extended to the “Tits-Alternative”, a property of linear groups established by J. Tits $[\mathbf {T}]$. 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 $[\mathbf {Th1}]$. This theory generalizes results known for pseudo-Anosov mapping classes $[\mathbf {FLP}]$.

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