Logarithmic growth of systole of arithmetic riemann surfaces along congruence subgroups

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Abstract

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 4/3-bound sysπ1(XPCH) ≥ 4/3 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume. © 2007 Applied Probability Trust.

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Katz, M. G., Schaps, M., & Vishne, U. (2007). Logarithmic growth of systole of arithmetic riemann surfaces along congruence subgroups. Journal of Differential Geometry, 76(3), 399–422. https://doi.org/10.4310/jdg/1180135693

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