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Zero entropy subgroups of mapping class groups

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Abstract

Given a group action on a surface with a finite invariant set we investigate how the algebraic properties of the induced group of permutations of that set affects the dynamical properties of the group. Our main result shows that in many circumstances if the induced permutation group is not solvable then among the homeomorphisms in the group there must be one with a pseudo-Anosov component. We formulate this in terms of the mapping class group relative to the finite set and show the stronger result that in many circumstances (e.g. if the surface has boundary) if this mapping class group has no elements with pseudo-Anosov components then it is itself solvable.

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Acknowledgments

The authors would like to thank Chris Leininger for reading an earlier draft of this paper and providing several useful comments and suggestions.

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Authors and Affiliations

  1. Northwestern University, Evanston, IL, 60208, USA

    John Franks

  2. Eastern Illinois University, Charleston, IL, USA

    Kamlesh Parwani

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  1. John Franks
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  2. Kamlesh Parwani
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Correspondence to Kamlesh Parwani.

Additional information

This work was partially supported by a grant from the Simons Foundation (#280151 to Kamlesh Parwani).

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Franks, J., Parwani, K. Zero entropy subgroups of mapping class groups. Geom Dedicata 186, 27–38 (2017). https://doi.org/10.1007/s10711-016-0178-9

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  • DOI: https://doi.org/10.1007/s10711-016-0178-9

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