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Systole growth for finite area hyperbolic surfaces. (English. French summary) Zbl 1295.30093

From the authors’ abstract: “We observe that the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature \((g,n)\) is greater than a function that grows logarithmically in terms of the ratio \(g/n\).”
Here of course \(g\) is the genus and \(n\) is the number of cusps. Also, \(2g-2+ n>0\). The paper’s main theorem is the following: For any \(g\geq 2\) and \(n\geq 1\),
\[ \text{sys}(g,n)> U\log({9\over n+1}), \] where \(U\) denotes the constant of Buser and Sarnak [Invent. Math. 117, No. 1, 27–56 (1994; Zbl 0814.14033)]. Also of interest is the following proposition: For all \((g,n)\) such that \(2g-2+n> 0\) and \(n\leq 2\), \(\text{sys}(g,n)< \text{sys}(g,n+1)\).

MSC:

30F10 Compact Riemann surfaces and uniformization

Citations:

Zbl 0814.14033
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References:

[1] Adams (C.).— Maximal cusps, collars , and systoles in hyperbolic surfaces. Indiana Math. J. 47, no. 2, p. 419-437 (1998). · Zbl 0912.53026
[2] Beardon (A.), Minda (D.).— The hyperbolic metric and geometric function theory. Quasiconformal mappings and their applications, p. 9-56, Narosa, New Delhi (2007). · Zbl 1208.30001
[3] Buser (P.), Sarnak (P.).— On the period matrix of a Riemann surface of large genus. With an appendix by J. H. Conway and N. J. A. Sloane. Invent. Math. 117, no. 1, p. 27-56 (1994). · Zbl 0814.14033
[4] Farb (B.), Margalit (D.).— A primer on mapping class groups. To appear in Princeton Mathematical Series. · Zbl 1245.57002
[5] Mumford (D.).— A remark on a Mahler’s compactness theorem. Proc. AMS 28, no. 1, p. 289-294 (1971). · Zbl 0215.23202
[6] Schmutz (P.).— Congruence subgroups and maximal Riemann surfaces. J. Geom. Anal. 4, p. 207-218 (1994). · Zbl 0804.32010
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