Florent Balacheff, Vincent Despré, Hugo Parlier
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.
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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.
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 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.
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.
We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that grows logarithmically in function of the ratio $g/n$.
We show that the length $R$ of a systole of a closed hyperbolic $n$-manifold $(n \geq 3)$ admitting a triangulation by $t$ $n$-simplices can be bounded below by a function of $n$ and $t$, namely \[ R \geq \frac{1}{2^{(nt)^{O(n^4t)} }} .\] We do this by finding a relation between the number of $n$-simplices and the diameter of the manifold and by giving explicit bounds for a well known relation between the length of the core curve of a Margulis tube and its radius. We prove the same result for finite volume manifolds, with a similar but slightly more involved proof.
We prove that for any \e>0, there exists a closed hyperbolic 4-manifold with a closed geodesic of length < \e.
Hyperbolic manifolds with a large number of systoles
本文对任意$n\geq 4$构造了紧双曲$n$-流形序列，证明了其systoles数量具有关于体积的下界，并改进了3维情形的界，将先前二维和三维的结果推广至更高维。
The main goal of this note is to show that the study of closed hyperbolic surfaces with maximum length systole is in fact the study of surfaces with maximum length homological systole. The same result is shown to be true for once-punctured surfaces, and is shown to fail for surfaces with a large number of cusps.
The systole of random hyperbolic 3-manifolds
研究Petri和Raimbault引入的随机双曲3-流形模型中systole的极限期望值，并给出闭式公式及数值近似。
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.
In this paper we study the systoles of arithmetic hyperbolic 2- and 3-manifolds. Our first result is the construction of infinitely many arithmetic hyperbolic 2- and 3-manifolds which are pairwise noncommensurable, all have the same systole, and whose volumes are explicitly bounded. Our second result fixes a positive number x and gives an upper bound for the least volume of an arithmetic hyperbolic 2- or 3-manifold whose systole is greater than x. We conclude by determining, for certain small values of x, the least volume of a principal arithmetic hyperbolic 2-manifold over Q or 3-manifold ove…
Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V. This extends a result of Adams and Reid, who showed that in the case that M is not hyperbolic, there is a universal bound of 7.35534... . As part of the proof, we establish a bound for the systole length of a non-compact finite volume hyperbolic manifold which grows asymptotically like (4/3)log(V).
We provide an explicit lower bound for the sytole in principal congruence covers of compact quaternionic hyperbolic manifolds. We also prove the optimality of this lower bound.
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$\log g$。
Density of systoles of hyperbolic manifolds
本文证明了闭双曲流形收缩极小测地线在正实数集中稠密，并揭示了其与Salem数及Salem猜想在算术双曲流形中的等价关系。
We find bounds for the length of the systole -- the shortest essential, non-peripheral closed curve -- for arithmetic punctured spheres with $n$ cusps, for $n=4$ through $n=12$, some of which were previously known due to Schmutz. This is shown using a correspondence between such surfaces and planar triangulations. We show that for $n=7,10,11$, arithmetic surfaces do not achieve the maximal systole length.
Systoles of hyperbolic hybrids
本文通过混合构造方法，得到各维度中具有任意小最短闭测地线且非拟算术的闭双曲流形，证明其与已知构造不可公度。
Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic fiel…
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…
Systole, inradius and rigidity of cusped hyperbolic 3-manifolds
建立了有限体积双曲三维流形中 systole 和内半径与体积之间的最优不等式，并确定了极值流形，包括八之字结补及其姐妹流形、Weeks-Matveev-Fomenko 流形等。
Maximal systole of hyperbolic surface with largest $S^3$ extendable abelian symmetry
本文给出了具有最大 $S^3$ 可扩展阿贝尔对称性的双曲曲面的最大 systole 公式，结果为 $2\mathrm{arccosh} K$，其中 $K$ 由 $L=4\cos^2(\pi/(g-1))$ 确定。
We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.
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 show that for every $n\geq 2$ and any $ε>0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $ε$. When $ε$ is sufficiently small these manifolds are non-arithmetic, and they are obtained by a generalised inbreeding construction which was first suggested by Agol for $n=4$. We also show that for $n\geq 3$ the volumes of these manifolds grow at least as $1/ε^{n-2}$ when $ε\to 0$.
Filling Links and Essential Systole
本文回答了Freedman和Krushkal的问题，在任意闭可定向3-流形中构造了具有大本质收缩的双曲填充链环，突破了先前对双曲链环的几何限制。
We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we show that, up to isometry, there are only finitely many hyperbolic structures on a surface that share a given ortho spectrum.
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible). There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient. It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contra…
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…
We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that the extremal surface is not conformally equivalent to the hyperbolic surface with maximal systole, yielding a first example of systolic extremality with this behavior.
On Systole, Kissing Number and Volume of Arithmetic Manifolds
本文通过超几何流形中系数和kissing数增长问题的解答，探讨算术群与算术流形的性质，总结二维结果及高维和其他局部对称空间的现状，并提出开放问题。
Any two geometric ideal triangulations of a cusped complete hyperbolic $3$-manifold $M$ are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check if two hyperbolic knots are equivalent, given geometric ideal triangulations of their complements. Given a geometric ideal triangulation of $M$, we also give a lower bound on the systole length of $M$ in terms of the number of tetrahedra and a lower …
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…
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…
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.
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。
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。
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.
We determine optimal inequalities for the systole of all hyperbolic compact surfaces of caracteristic -1. First, we study the geometry and topology of these surfaces. Then, we describe the action of modular groups on Teichmüller spaces. Finaly, we give cell decompositions of fundamental domains such as the set of systoles is constant over a cell. Other optimal inequalities for other metric invariants are also given.
We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of primitive closed geodesics with length in a given interval that are uniform for all closed hyperbolic manifolds with bounded geometry. The proofs rely on the Selberg trace formula.
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…
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。
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 …
Hyperbolic manifolds without positive spun triangulations
研究由闭双曲三维流形和简单闭测地线组成的对，利用Choi结果给出无前述正旋转理想三角剖分的例子，如Vol3及相关测地线，为Vol3无此类三角剖分猜想提供证据。
This article is devoted to the variational study of two functions defined over some Teichmueller spaces of hyperbolic surfaces. One is the systole of geodesic loops based at some fixed point, and the other one is the systole of arcs.\par For each of them we determine all the critical points. It appears that the systole of arcs is a topological Morse function, whereas the systole of geodesic loops have some degenerate critical points. However, these degenerate critical points are in some sense the obvious one, and they do not interfere in the variational study of the function.\par At a nondegen…
Shape of filling-systole subspace in surface moduli space and critical points of systole function
本文研究曲面模空间中填充systole子空间$X_g$的形状，证明其半径$r$邻域无法覆盖厚部，并构造距离小和大的systole函数临界点。
Counting and entropy for hyperbolic surface amalgams
研究闭合双曲面拼接中闭合测地线数量的增长，证明拓扑熵与体积熵相等，但其行为受几何数据影响，特别是通过系统孔径长度和拼接曲线长度的上下界影响，熵在无系统孔径下界时可指数增长。
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。
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。
We study the Weyl chamber length boundary both of the Hitchin and of the maximal character varieties and determine therein an open set of discontinuity for the action of the mapping class group. This result is obtained as consequence of a canonical decomposition of a geodesic current on a surface of finite type arising from a topological decomposition of the surface along special geodesics. We show that each component either is associated to a measured lamination or has positive systole. For a current with positive systole, we show that the intersection function on the set of closed curves is …
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。
Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we establish an upper bound of the length of an $n^{th}$ shortest closed geodesic as a logarithmic function of $V$ in $M\setminus L$. Our works complement the work of Lakeland and Leininger \cite{Christopher} on the upper bound of systole length.
We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author's result to the with-boundary setting. In the second part of the paper we comment on another direction for extending this result, via the systole of loops function.
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。
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。
In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed geodesic on the surface. This will follow from a more general bound applying to any family of hyperbolic surfaces which admits a Bers' constant and with a lower bound on systole length.
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。
Geometric and arithmetic properties of Löbell polyhedra
本文研究Löbell多面体的几何与算术性质，证明其短闭测地线可任意小且在可公度意义下仍为无限族，并精确刻画其中的拟算术成员。
In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surface. We show that the spine systole is a proper function and has its global minima precisely on the extremal surfaces (those containing the biggest possible discs). We also study minimal spines, which are critical points for the length functional. We completely classify minimal s…
In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.
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