Abstract
A closed hyperbolic Riemann surface \(M\) is said to be \(K\)-quasiconformally homogeneous if there exists a transitive family \(\fancyscript{F}\) of \(K\)-quasiconformal homeomorphisms. Further, if all \([f] \subset \fancyscript{F}\) act trivially on \(H_1(M;\mathbb {Z})\), we say \(M\) is Torelli-\(K\)-quasiconformally homogeneous. We prove the existence of a uniform lower bound on \(K\) for Torelli-\(K\)-quasiconformally homogeneous Riemann surfaces. This is a special case of the open problem of the existence of a lower bound on \(K\) for (in general non-Torelli) \(K\)-quasiconformally homogeneous Riemann surfaces.
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Notes
The result by Nicholas Vlamis is obtained in “Quasiconformal homogeneity and subgroups of the mapping class group”, available at arXiv:1309.7026.
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Acknowledgments
The author wishes to thank Professor Vladimir Markovic for providing an interesting project and for his effective mentoring. This work was done under a Ryser Summer Undergraduate Research Fellowship at Caltech.
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Greenfield, M. A lower bound for Torelli-\(K\)-quasiconformal homogeneity. Geom Dedicata 177, 61–70 (2015). https://doi.org/10.1007/s10711-014-9977-z
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DOI: https://doi.org/10.1007/s10711-014-9977-z