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 sysπ 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 4/3bound sysπ 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.

Locations

• Journal of Differential Geometry - View - PDF
• arXiv (Cornell University) - View - PDF