Hugo Parlier, Yunhui Wu, Yuhao Xue
In this note we show that the expected value of the separating systole of a random surface of genus $g$ with respect to Weil-Petersson volume behaves like $2\log g $ as the genus goes to infinity. This is in strong contrast to the behavior of the expected value of the systole which, by results of Mirzakhani and Petri, is independent of genus.
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In this article, we provide bounds on systoles associated to a holomorphic $1$-form $ω$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,ω)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $ω$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.
We obtain the exact values of the systoles of these hyperbolic surfaces of genus $g$ with cyclic symmetries of the maximum order and the next maximum order. Precisely: for genus $g$ hyperbolic surface with order $4g+2$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+\cos \fracπ{2g+1}+\cos \frac{2π}{2g+1})$ when $g\ge 7$, and for genus $g$ hyperbolic surface with order $4g$ cyclic symmetry, the systole is $2\mathrm{arccosh} (1+2\cos \fracπ{2g})$ when $g\ge4$.
We present a construction of sequences of closed hyperbolic surfaces that have long systoles which form pants decompositions of these surfaces. The length of the systoles of these surfaces grows logarithmically as a function of their genus.
Non-simple systoles on random hyperbolic surfaces for large genus
本文研究了Weil-Petersson测度下亏格为g的黎曼曲面模空间$\mathcal{M}_g$上随机双曲曲面的非简单收缩的渐近行为，证明当g→∞时其长度渐近于$\log g$。
In this article we explore the relationship between the systole and the diameter of closed hyperbolic orientable surfaces. We show that they satisfy a certain inequality, which can be used to deduce that their ratio has a (genus dependent) upper bound.
We study the number and the length of systoles on complete finite area orientable hyperbolic surfaces. In particular, we prove upper bounds on the number of systoles that a surface can have (the so-called kissing number for hyperbolic surfaces). Our main result is a bound which only depends on the topology of the surface and which grows subquadratically in the genus.
The purpose this article is to try to understand the mysterious coincidence between the asymptotic behavior of the volumes of the Moduli Space of closed hyperbolic surfaces of genus $g$ with respect to the Weil-Petersson metric and the asymptotic behavior of the number of arithmetic closed hyperbolic surfaces of genus $g$. If the set of arithmetic surfaces is well distributed then its image for any interesting function should be well distributed too. We investigate the distribution of the function systole. We give several results indicating that the systoles of arithmetic surfaces can not be c…
Small genus, small index critical points of the systole function
本文计算了Teichmüller空间上systole函数一族临界点的指标，该族的存在暗示Thurston脊柱中systole不决定同调基的层；并研究了最小填充集，确定了亏格5中基数为8的例子及正则直角m边形镶嵌相关的最小填充集的基数范围，使用整数线性规划和对称性破坏技术。
We prove that the extremal length systole of genus two surfaces attains a strict local maximum at the Bolza surface, where it takes the value $\sqrt{2}$.
Large genus asymptotics for lengths of separating closed geodesics on random surfaces
本文研究了模空间上关于 Weil-Petersson 测度的随机双曲曲面几何量，证明了当亏格趋于无穷时分离 systole、半领子宽度和最短分离闭多重测地线长度的渐近行为，并讨论了相关极值量的渐近性质。
We construct infinite families of closed hyperbolic surfaces that are local maxima for the systole function on their respective moduli spaces. The systole takes values along a linearly divergent sequence $(L_n)_{n\geq 1}$ at these local maxima. The only surface corresponding to $L_1\approx 3.057$ is the Bolza surface in genus $2$. For every genus $g\geq 13$, we obtain either one or two local maxima in $\mathcal{M}_g$ whose systoles have length $L_2\approx 5.909$. For each $n\geq 3$, there is an arithmetic sequence of genera $(g_k)_{k\geq 1}$ such that the number of local maxima of the systole …
For any $\varepsilon>0$, we construct a closed hyperbolic surface of genus $g=g(\varepsilon)$ with a set of at most $\varepsilon g$ systoles that fill, meaning that each component of the complement of their union is contractible. This surface is also a critical point of index at most $\varepsilon g$ for the systole function, disproving the lower bound of $2g-1$ posited by Schmutz Schaller.
The so-called {\it kissing number} for hyperbolic surfaces is the maximum number of homotopically distinct systoles a surface of given genus $g$ can have. These numbers, first studied (and named) by Schmutz Schaller by analogy with lattice sphere packings, are known to grow, as a function of genus, at least like $g^{\sfrac{4}{3}-ε}$ for any $ε>0$. The first goal of this article is to give upper bounds on these numbers; in particular the growth is shown to be sub-quadratic. In the second part, a construction of (non hyperbolic) surfaces with roughly $g^{\sfrac{3}{2}}$ systoles is given.
We investigate the geometry of $π_1$-injective surfaces in closed hyperbolic 3-manifolds. First we prove that for any $e>0$, if the manifold $M$ has sufficiently large systole $\sys_1(M)$, the genus of any such surface in $M$ is bounded below by $\exp((1/2-e)\sys_1(M))$. Using this result we show, in particular, that for congruence covers $M_i\to M$ of a compact arithmetic hyperbolic 3-manifold we have: (a) the minimal genus of $π_1$-injective surfaces satisfies $\log \sysg(M_i) \gtrsim (1/3)\log\vol(M_i)$; (b) there exist such sequences with the ratio Heegard genus$(M_i)/\sysg(M_i) \gtrsim \v…
The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces. In this article we provide a new construction for a sequence of surfaces with systoles that grow logarithmically in their genera. We do this by combining a construction for graphs of large girth and a count of the number of $\mathrm{SL}_2(\mathbb{Z})$ matrices with positive entries and bounded trace…
We study the systole of a random surface, where by a random surface we mean a surface constructed by randomly gluing together an even number of triangles. We study two types of metrics on these surfaces, the first one coming from using ideal hyperbolic triangles and the second one using triangles that carry a given Riemannian metric. In the hyperbolic case we compute the limit of the expected value of the systole when the number of triangles goes to infinity (approximately 2.484). We also determine the asymptotic probability distribution of the number of curves of any finite length. This turns…
For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation between (locally) maximal values of the systole function and the number of shortest saddle connections.
Random surfaces with large systoles
提出两种受图论启发的构造，用于生成面积增长时短程线对数增长的随机曲面序列，还证明了给定亏格的闭可定向双曲曲面最大短程线的新下界。
Every closed surface of genus at least 18 is Loewner
本文改进了Riemannian曲面体积熵的上界，证明了亏格至少为18的可定向闭曲面及亏格至少为11的非正曲率闭曲面均满足Loewner的systolic比不等式。
In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze the growth of the genera of minimal surfaces across commensurability classes. These results have applications to the study of how Heegard genus grows across commensurability classes.
In this article we study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfaces of genus $g$ is uniformly comparable to $\frac{1}{g^2}$ in $g$. In the proof of the upper bound, for any constant $ε>0$, we will construct a closed Riemann surface of genus $g$ in any $ε$-thick part of moduli space such that it admits a pants decomposition whose boundary curves all have length equal to $ε$, a…
In this article we show that for every finite area hyperbolic surface $X$ of type $(g,n)$ and any harmonic Beltrami differential $μ$ on $X$, then the magnitude of $μ$ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil-Petersson norm of $μ$ over the square root of the systole of $X$ up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil-Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the…
Given a hyperelliptic Klein surface, we construct companion Klein bottles, extending our technique of companion tori already exploited by the authors in the genus 2 case. Bavard's short loops on such companion surfaces are studied in relation to the original surface so to improve a systolic inequality of Gromov's. A basic idea is to use length bounds for loops on a companion Klein bottle, and then analyze how curves transplant to the original nonorientable surface. We exploit the real structure on the orientable double cover by applying the coarea inequality to the distance function from the r…
A note on surfaces with large systoles
该注记通过延续前期有限覆盖直径研究，采用常扭转裤形分解框架，证明亏格足够大时存在闭双曲曲面，其systole下界改进为log g级，突破此前2/9的界，证明由GPT-5.6 Sol完成。
Linear programming bounds for hyperbolic surfaces
本研究通过线性规划方法为双曲面提供了新的上界和下界，涉及系绳长、kissing 数和谱特性，适用于低亏格和高亏格情况。
Small Systle Sets and Coxeter Groups
本文基于 Coxeter 群理论与初等数论，改进 Schmutz 的方法，构造了具有任意大亏格且其 systole 集合基数低至 o(g/ ln g) 仍能填充曲面的闭双曲曲面。
Finite covers of a product of surfaces with bounded rank and arbitrarily large systole
本文构造了亏格二双曲曲面乘积的有限覆盖，其基本群生成元数有界而内射半径趋于无穷，为高秩对称空间上的 Avramidi-Delzant 猜想提供了反例。
Intersection Number, Length, and Systole on Compact Hyperbolic Surfaces
本研究针对紧致双曲曲面的相互作用强度I(X)，分别推导了其在模空间$M_g$中曲面趋于无穷时依收缩线长度的渐近行为，以及亏格g趋于无穷时$M_g$上I(X)最小值的近似行为。
We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the large genus limit of the expected systole.
We give asymptotic bounds for the optimal Lipschitz constants for the systole map from the Teichmuller space to the curve complex. We give similar results to those known for closed surfaces in the cases when the genus is fixed or the ratio of genus and punctures is a rational number.
A proof that the separating curve complex of the closed genus two surface has a quasi-distance formula and is delta hyperbolic using tools of Masur and Schleimer. This answers in the affirmative a Conjecture of Schleimer.
Trace Systoles and Sink Constant
该研究定义了曲面基本群到SL(2,ℂ)表示的迹收缩长度，借助Markoff映射对应关系，显式算出三类曲面相对特征簇上的最优界，并得到双曲流形收缩不等式与非Fuchsian表示的相关推论。
Systole functions and Weil-Petersson geometry
本文研究Teichmüller空间中沿systolic曲线的测地线长度函数梯度的L^p范数，证明其与systole的幂次一致可比，并由此得出最小Weil-Petersson全纯截面曲率有统一负上界，否定了Mirzakhani的问题。
Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the lengths of homologically independent curves on closed Riemannian surfaces. As a consequence, we show that for any $λ\in (0,1)$ there exists a constant $C_λ$ such that every closed Riemannian surface of genus $g$ whose area is normalized at $4π(g-1)$ has at least $[λg]$ homologically independent loops of length at most $C_λ\log(g)$. This result extends Gromov's asy…
The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number of appearances of fixed finite sets of combinatorial types of curves. Of any such set we determine the asymptotics of the probability distribution. It turns out that these distributions are independent of the genus in an appropriate sense. As an application of our results we stu…
In this article we find an upper and lower bound for the slope of genus g hyperelliptic Lefschetz fibrations, which is sharp when g = 2, and demonstrate the strong connection, in general, between the slope of hyperelliptic genus g Lefschetz fibrations and the number of separating vanishing cycles. Specifically, we show that the slope is greater than 4-4/g if and only if the fibration contains separating vanishing cycles. We also improve the existing bound on s/n, the ratio of number of separating vanishing cycles to the number of non-separating vanishing cycles, for hyperelliptic Lefschetz fib…
Asymptotics of shortest filling closed multi-geodesics
本文研究了双曲面中闭合多测地线的渐进行为，证明了其长度与亏格和测地线长度的对数相关，并展示了在随机双曲面模型中的应用。
For each closed surface of genus $g\ge3$, we find a finite subcomplex of the separating curve complex that is rigid with respect to incidence-preserving maps.
We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicative constant in Gromov's theorem. We show that every surface of genus at least 20 is Loewner. Finally, we relate, in higher dimension, the isoembolic ratio to the minimal entropy.
Reducible handle additions along separating slopes
本文证明在紧致可定向简单3-流形边界上，若两个分离坡道对应的柄添加均为可约，则它们之间的距离为零，从而在任意边界亏格下验证了Qiu-Zhang猜想中可约-可约情形。
A note on reducing spheres for the genus-4 Heegaard surface in the 3-sphere
本文提出一个充分条件，用于确定非分离弱减少对被简化球分离的情况，并将简化球复形中的连通性问题转化为证明任意两个顶点在复形中连通的问题。
For a handlebody of genus $g\geq6$ it is shown that every automorphism of the complex of separating meridians can be extended to an automorphism on the complex of all meridians and, in consequence, it is geometric.
Computing the second and third systoles of a combinatorial surface
本文研究了组合曲面上加权图中第二和第三短同伦非平凡闭合游走的计算问题，通过分析最短同伦非平凡曲线的构型，给出了时间复杂度为 $O(n^2\log n)$ 和 $O(n^3)$ 的算法，并在亏格与边界数固定时将第二短游走计算优化至 $O(n\log n)$。
We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.
We prove that the separating curve graph of a connected, compact, orientable surface with genus at least 3 and a single boundary component is not relatively hyperbolic. This completes the classification of when the separating curve graph is hyperbolic and relatively hyperbolic initiated by previous works of the authors.
Finite rigid sets of the non-separating curve complex
本文证明了有限类型表面（亏格≥3）的非分离曲线复形可由有限刚集穷尽。
In this paper we investigate the systolic landscape of translation surfaces for fixed genus and fixed angles of their cone points. We furthermore study how the systoles of a translation surface relate to the systoles of its graph of saddle connections. This allows us to develop an algorithm to compute the systolic ratio of origamis in the stratum $\mathcal{H}(1,1)$. We compute the maximal systolic ratio of all origamis in $\mathcal{H}(1,1)$ with up to 67 squares. These computations support a conjecture of Judge and Parlier about the maximal systolic ratio in $\mathcal{H}(1,1)$.
In this note we find new relations in the mapping class group of a genus two surface with n boundary components for n=1,..., 8 which induce a genus two Lefschetz fibration $CP^2#13CP^2bar \to S^2$ with n disjoint sections. As a consequence, we observe any holomorphic genus 2 Lefschetz fibration without separating singular fibers admits a section.
We study filling sets of simple closed curves on punctured surfaces. In particular we study lower bounds on the cardinality of sets of curves that fill and that pairwise intersect at most k times on surfaces with given genus and number of punctures. We are able to establish orders of growth for even k and show that for odd k the orders of growth behave differently. We also study the corresponding questions when one requires that the curves be represented as systoles on hyperbolic complete finite area surfaces.
In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(g\geq 2)$, the first eigenvalue of $X_g$ is greater than $\frac{\mathcal{L}_1(X_g)}{g^2}$ up to a uniform positive constant multiplication. Where $\mathcal{L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $g \to \infty$.
Entropy and self-intersection number of geodesic currents on compact hyperbolic surfaces
研究测地电流在双曲面上的熵与自交数关系，建立熵与自交数及系统最小长度的定量上界，证明小自交数导致小熵。
We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are $O(1/g)$, where $g$ is the genus of the surface.
A non-separating multicurve of a surface S of genus g with m punctures is a multicurve c so that S-c is connected. For k>0 define the graph of non-separting k-multicurves to be the graph whose vertices are non-separating multicurves with k components and where two such multicurves are connected by an edge if they can be realized disjointly. We show that if k is smaller than g/2+1 then this graph is hyperbolic.
Let $M$ be a simple 3-manifold such that one component of $\partial M$, say $F$, has genus at least two. For a slope $α$ on $F$, we denote by $M(α)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $α$ on $F$. If $M(α)$ is reducible, then $α$ is called a reducing slope. In this paper, we shall prove that the distance between two separating, reducing slopes on $F$ is at most 4.
We construct a small, hyperbolic 3-manifold $M$ such that, for any integer $g\geq 2$, there are infinitely many separating slopes $r$ in $\partial M$ so that $M(r)$, the 3-manifold obtained by attaching a 2-handle to $M$ along $r$, is hyperbolic and contains an essential separating closed surface of genus $g$. The result contrasts sharply with those known finiteness results on Dehn filling, and it also contrasts sharply with the known finiteness result on handle addition for the cases $g=0,1$. Our 3-manifold $M$ is the complement of a hyperbolic, small knot in a handlebody of genus 3.
Mirzakhani's frequencies of simple closed geodesics on hyperbolic surfaces in large genus and with many cusps
本文证明了Delecroix等人的猜想，给出了亏格为$g$且带$n$个尖点的闭双曲曲面上分离与非分离简单闭测地线频率之比的亏格渐近行为，并明确给出函数$f(\ rac{n}{g})$，揭示了频率的拓扑本质。
Holomorphic Curves in Moduli Spaces Are Quasi-Isometrically Immersed
该论文证明模空间中全纯曲线在边缘单值为无限阶时是拟等距浸入，并在附加条件下嵌入Teichmüller空间，同时改进了Parshin-Arakelov有限性定理。
Suppose M is a compact orientable irreducible 3-manifold with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a possibly stabilized copy of P or the Hempel distance of the splitting P is no greater than twice the genus of Q. More generally, if P and Q are bicompressible but weakly incompressible connected closed separating surfaces in M then either a) P and Q can be well-separated or b) P and Q are isotopic or c) the Hempel distance of P is no greater than twice the genus of Q.
We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove that the level 2 twist subgroup is normally generated in the mapping class group by a crosscap pushing map along a non-separating two-sided simple loop for genus $g\geq 5$ and $g=3$. As an application, we calculate the first homology group of the level 2 twist subgroup for genus $…
We prove that the separated curve complex of a closed orientable surface of genus g is (g-3)-connected. We also obtain a connectivity property for a separated curve complex of the open surface that is obtained by removing a finite set from a closed one, but it is then assumed that the removed set is endowed with a partition and that the separating curves respect that partition. These connectivity statements have implications for the algebraic topology of the moduli space of curves.
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