LOGARITHMIC GROWTH OF SYSTOLE OF ARITHMETIC RIEMANN SURFACES ALONG CONGRUENCE SUBGROUPS

By Mikhail G. Katz, M. Schaps, U. Vishne. 2005

92 citations recorded by the source corpus.

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 sysπ1(XPCH) ≥ 4 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f7889d620081867495330b37b39499456111cd43

Citations in the corpus

• Homological Invariance For Asymptotic Invariants and Systolic Inequalities (2007, unknown): “A discussion of this optimality result may be found in [ KSV07 ].”