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Logarithmic growth of systole of arithmetic riemann surfaces along arithmetic riemann surfaces along

Mikhail G. Katz Bar Ilan University Mary Schaps Bar Ilan University Uzi Vishne Bar Ilan University

Differential Geometry mathscidoc:1609.10026

Journal of Differential Geometry, 76, (3), 399-422, 2007
We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole,and denoted sysπ1. P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3bound sysπ1(XPCH) ≥ 43 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.
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@inproceedings{mikhail2007logarithmic,
  title={LOGARITHMIC GROWTH OF SYSTOLE OF ARITHMETIC RIEMANN SURFACES ALONG ARITHMETIC RIEMANN SURFACES ALONG},
  author={Mikhail G. Katz, Mary Schaps, and Uzi Vishne},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908202735261014683},
  booktitle={Journal of Differential Geometry},
  volume={76},
  number={3},
  pages={399-422},
  year={2007},
}
Mikhail G. Katz, Mary Schaps, and Uzi Vishne. LOGARITHMIC GROWTH OF SYSTOLE OF ARITHMETIC RIEMANN SURFACES ALONG ARITHMETIC RIEMANN SURFACES ALONG. 2007. Vol. 76. In Journal of Differential Geometry. pp.399-422. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908202735261014683.
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