标题： 关于大系统长的黎曼曲面的一篇注记 显示英文标题

标题： A Note on Riemann Surfaces of Large Systole

摘要： 我们研究大系统问题，这涉及紧致双曲黎曼曲面，其系统长度（最短不可缩回环的长度）在 genus 中以对数方式增长。 Buser 和 Sarnak 的一种构造的推广由 Katz、Schaps 和 Vishne 完成，该构造使用固定共闭算术 Fuchsian 的主“同余”子群，在当前已知的增长常数中达到了\gamma = 4 /3。 我们证明，在同余情况下，使用算术 Fuchsian 的这种构造，\gamma 的最佳可能值为该值。 最后一节将大系统问题与正则图的类似大周长问题进行了比较。 显示英文摘要

摘要： 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 \gamma = 4/3. We prove that this is the best possible value of \gamma 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.