Journal of the Ramanujan Mathematical Society

Volume 28, Issue 3, September 2013 pp. 359–377.

A note on Riemann surfaces of large systole

Authors: Shotaro Makisumi
Author institution: Department of Mathematics, Stanford University, Stanford, CA 94305, USA

Summary: We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and Sarnak by Katz, Schaps, and Vishne, which uses principal “congruence” subgroups of a fixed cocompact arithmetic Fuchsian, achieves the current maximum known growth constant of γ = 4/3. We prove that this is the best possible value of γ for this construction using arithmetic Fuchsians in the congruence case. The final section compares the large systole problem with the analogous large girth problem for regular graphs.

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