Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups

TL;DR

This paper demonstrates that the systole length of certain arithmetic Riemann surfaces grows logarithmically with genus, extending previous results and providing new bounds using quaternion algebra techniques.

Contribution

It generalizes the logarithmic systole growth results to principal congruence subgroups of arbitrary arithmetic surfaces using novel trace estimates.

Findings

Logarithmic growth of systole in genus for arithmetic surfaces

Sharp 4/3-bound for Hurwitz surface towers

Extension of systole bounds to hyperbolic 3-manifolds

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.