Construction of hyperbolic Riemann surfaces with large systoles Hugo Akrout and Bjoern Muetzel June 6, 2018 Abstract Let S be a compact hyperbolic Riemann surface of genus g 2. We call a systole a shortest simple closed geodesic in S and denote by sys(S) its length.≥ Let msys(g) be the maximal value that sys( ) can attain among the compact Riemann surfaces of genus g. We · call a (globally) maximal surface Smax a compact Riemann surface of genus g whose systole has length msys(g). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating msys( ) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact· hyperbolic Riemann surfaces with cusps. Keywords: Riemann surfaces, systoles, intersystolic inequalities, maximal surfaces. Mathematics Subject Classification (2010): 30F10, 32G15 and 53C22. 1 Introduction Let S be a compact hyperbolic Riemann surface of genus g 2. A systole of S is a shortest ≥ simple closed geodesic. We denote by sys(S) its length. Let msys(g) be the value msys(g) = sup sys(S) S compact hyperbolic Riemann surface of genus g 2 . { | ≥ } Due to Mumford’s generalization of Mahler’s compactness theorem in [Mu], this supremum is a maximum. The exact value of msys(g) is only known for g = 2. We also have the following estimates from [BS]: There exists a universal, but unknown, constant C > 0 such that for all genera g 2 ≥ arXiv:1305.5510v7 [math.DG] 15 Aug 2016 C log(g) msys(g) 2 log(4g 2). (1) · ≤ ≤ − Here the upper bound follows from a simple area argument (see [BS]). The first account of a lower bound is due to Buser in [Bu1]. Here an infinite sequence of surfaces with lower bound of order log(g) is constructed. Using arithmetic surfaces it was then shown by Buser and Sarnak in [BS] that there is a infinite sequence of genera (g ) with p k k 4 msys(gk) log(gk) c ≥ 3 − 0 where c0 is constant. This construction was then generalized in [KSV1], [KSV2] and [KKSV], where more families of compact hyperbolic Riemann surfaces satisfying the above inequality 4 can be found. Asymptotically the factor 3 is the best known to date and Makisumi (see [Ma], 1 Theorem 1.6) showed that, in some sense, this lower bound is optimal for the generalized Buser-Sarnak construction. The study of surfaces whose systole length is a global or local maximum in the moduli space g of compact hyperbolic Riemann surfaces of genus g 2 was initiated by Schmutz (see [Sc1], M ≥ [Sc2] and [Sc3]). Here he also provides a number of interesting properties of these surfaces. The characterization of maximal surfaces was continued in [Ba], [Ak], [Ge] and [Pa1]. Here it was shown that A (locally) maximal surface of genus g has at least 6g 5 systoles [Sc2], [Ba]. • − There is only a finite number of maximal surfaces of genus g [Sc2], [Ba]. • The systole function sys( ) is a topological Morse function on the moduli space [Ak]. • · All systoles of maximal surfaces are non-separating (see [Pa1], Claim on p. 336). • An open question is, whether msys( ) is a monotonously increasing function with respect to the · genus. Though we can not prove or disprove this result, we can at least show the following intersystolic inequalities: Theorem 1.1. Let msys(g) be the maximal value that sys( ) can attain among the compact · hyperbolic Riemann surfaces of genus g 2. ≥ 1. msys(k(g 1) + 1) > msys(g) for k N 0, 1 . − ∈ \{ } msys(g) 2. msys(g + 1) > 2 . 3. If msys(g ) msys(g ), then msys(g + g 1) > min msys(g2) , msys(g ) . 2 ≥ 1 1 2 − { 2 1 } We note that Theorem 1.1-1 without a sharp inequality can be obtained by constructing a normal cover of a maximal surface in a similar fashion (see Note 2.2). This is due to the well known fact that the injectivity radius does not decrease in a normal cover of a closed Riemannian surface. The theorem is obtained by cutting and pasting maximal surfaces to construct compact hyper- bolic Riemann surfaces with large systoles. Here the main tool is Lemma 2.1, a collar lemma for systoles. As a result we obtain from Theorem 1.1-1 or the covering construction: If S is a compact Riemann surface of genus g, such that sys(S) 4 log(g) c , then for l = k (g 1) + 1 ≥ 3 − 0 · − 4 4 4 msys(l) log(l) log(k)+ c = log(l) c(k) for all k g. ≥ 3 − 3 0 3 − ≪ The concrete examples of genera g for which msys(g) 4 log(g) c is shown are sparse. The ≥ 3 − 0 above inequality, however, suggests that at least a slightly lower bound holds for a large number of genera g. Furthermore, by construction, we obtain a continuous parameter family (St) ∈ − 1 1 k of compact t ( 2 , 2 ] hyperbolic Riemann surfaces of genus k(g 1) + 1, such that − sys(St) = msys(g). 2 This shows that though the systoles of these surfaces are large, none of these surfaces can be maximal, as this would be a contradiction to the finiteness of the number of these surfaces. In the following table Tab. 1 we give a summary of compact hyperbolic Riemann surfaces of genus g 25 with maximal known systoles thus providing a reference and benchmark for further ≤ studies. Most of these are constructed using the examples presented in [Ca],[KSV1], [Sc2] and [Sc3] by applying Theorem 1.1 or the covering argument. genus g surface (name and/or (systole length)/ systole 2 log(4g 2) reference − constructed from) log(g) length 2 M ∗ 4.41 3.06 3.58 [Sc2](Bolza) 3 M(3) 3.63 3.98 4.61 [Sc2] (Wiman) 4 M(4) 3.34 4.62 5.28 [Sc2] 5 S5 3.05 4.91 5.78 [Sc3] 6 I6 2.85 5.11 6.18 [Ca] 7 H7 (via PSL(2,8)) 2.98 5.80 6.52 [KSV1] 8 7 M ∗ 1.47 3.06 6.80 normal cover × 9 2 S 2.24 4.91 7.05 normal cover × 5 10 3 M(4) 2.01 4.62 7.28 normal cover × 11 I(x z) 2.49 5.98 7.48 [Sc2] | ∗ 12 11 M 1.23 3.06 7.66 normal cover × 13 M (via 2 H ) 2.26 5.80 7.82 normal cover 13 × 7 14 H14 (via PSL(2, 13)) 2.61 6.89 7.98 [KSV1] 15 7 M(3) 1.47 3.98 8.12 normal cover × 16 3 I6 1.84 5.11 8.25 normal cover ×3 17 H17 (via (C2) .PSL(2, 7)) 2.69 7.61 8.38 [KSV1] 18 via H17 1.32 3.80 8.50 Th. 1.1-2 19 M (via 3 H ) 1.97 5.80 8.61 normal cover 19 × 7 20 H17, M(4) 1.27 3.80 8.71 Th. 1.1-3 21 2 I(x z) 1.96 5.98 8.81 normal cover × | 22 7 M(4) 1.50 4.62 8.91 normal cover × 23 B1 2.04 6.39 9.00 [Sc2] 24 via B1 1.01 3.19 9.09 Th. 1.1-2 25 2 M 1.80 5.80 9.17 normal cover × 13 Table 1: Compact hyperbolic Riemann surfaces with large systoles of genus g 25. ≤ 3 It follows furthermore from the known examples in genus 2 and 3 and Theorem 1.1-1 or Note 2.2: Let msys(g) be the maximal value that sys( ) can attain among the compact hyperbolic · Riemann surfaces of genus g 2. Then ≥ 3.06 msys(2) < msys(g) for all g and 3.98 msys(3) < msys(g) if g odd. (2) ≃ ≤ In Section 3 we use the same methods to derive similar inequalities for non-compact hyperbolic surfaces with cusps (see [Bu2], Example 1.6.8 for an exact definition). Let S be a hyperbolic Riemann surface of signature (g,n), i.e. of genus g and with n cusps. Let msys(g,n) be the value msys(g,n) = sup sys(S) S non-compact hyperbolic Riemann surface of signature (g,n) . { | } As in the case of the compact hyperbolic surfaces this supremum is a maximum due to Mahler’s compactness theorem. The known bounds for the value of msys(g,n) are the following: For (g,n) = (0, 3) and n 2 6 ≥ 12g 12 + 6n 2 arcsinh(1) msys(g,n) 4 log − . (3) ≤ ≤ n The upper bound follows again from an area argument (see [Sc1]), whereas the lower bound is due to the collar lemma for Riemann surfaces (see [Bu2], Theorem 4.1.1). More refined estimates have been obtained recently in [FP]. Heuristically speaking, if we fix the genus g and increase the number n of cusps continuously, then some of the cusps must move closer together and there will be a simple closed geodesic of bounded length surrounding two or several cusps. Such a geodesic is separating the surface into two parts. Indeed it can be shown that: Let Smax be a maximal hyperbolic Riemann surface of signature (g,n), where n 25g. • ≥ Then all systoles of Smax are separating. ([Pa1], Proposition 3.1). This will be important in the statement of the following theorem. Theorem 1.2. Let Smax be a maximal non-compact hyperbolic Riemann surface of signature (g,n) = (0, 4), where 3g 3+ n > 0 and n 2 that has a separating systole.