Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups
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.
Keywords
Cite
@article{arxiv.math/0505007,
title = {Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups},
author = {Mikhail G. Katz and Mary Schaps and Uzi Vishne},
journal= {arXiv preprint arXiv:math/0505007},
year = {2007}
}
Comments
25 pages. To appear in Journal of Differential Geometry