Logarithmic Growth of Systole of Arithmetic Riemann Surfaces along Congruence Subgroups
نویسنده
چکیده
P. Buser and P. Sarnak constructed Riemann surfaces whose systole behaves logarithmically in the genus. 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 Hurwitz surfaces. Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.
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