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On the displacement of generators of free Fuchsian groups

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Abstract

We prove an inequality that must be satisfied by displacement of generators of free Fuchsian groups, which is the two-dimensional version of the \(\log (2k-1)\) Theorem for Kleinian groups due to Anderson et al. (J Differ Geom 44:738–782, 1996). As applications, we obtain quantitative results on the geometry of hyperbolic surfaces such as the two-dimensional Margulis constant and lengths of a pair of based loops, which improves a result of Buser’s.

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Acknowledgements

I would like to thank Peter Shalen for his invaluable advice, gracious support and encouragement throughout this project. I thank David Dumas for pointing out the Proof of Lemma 4.4, Danny Calegari for helpful conversations and Yong Hou for commenting on the draft.

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  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Yan Mary He

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  1. Yan Mary He
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Correspondence to Yan Mary He.

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He, Y.M. On the displacement of generators of free Fuchsian groups. Geom Dedicata 200, 255–264 (2019). https://doi.org/10.1007/s10711-018-0369-7

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  • DOI: https://doi.org/10.1007/s10711-018-0369-7

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