TY - JOUR

T1 - Logarithmic growth of systole of arithmetic riemann surfaces along congruence subgroups

AU - Katz, Mikhail G.

AU - Schaps, Mary

AU - Vishne, Uzi

PY - 2007

Y1 - 2007

N2 - 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 sysπ1. 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 sysπ1(XPCH) ≥ 4/3 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.

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

UR - http://www.scopus.com/inward/record.url?scp=34547260501&partnerID=8YFLogxK

U2 - 10.4310/jdg/1180135693

DO - 10.4310/jdg/1180135693

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AN - SCOPUS:34547260501

SN - 0022-040X

VL - 76

SP - 399

EP - 422

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

IS - 3

ER -