Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups

Mikhail G. Katz; Mary Schaps; Uzi Vishne

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Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups

Mikhail G. Katz·Mary Schaps·Uzi Vishne

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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 syspi_1. 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 syspi_1(X_{\PCH}) > 4/3 \log(g(X_{\PCH})). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

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BibTeX RIS

Mikhail G. Katz, Mary Schaps, Uzi Vishne. 2007-02-07. Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. https://arxiv.org/abs/math/0505007

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