Genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume

The genus spectrum of a hyperbolic 3-manifold

In this article we study the spectrum of totally geodesic surfaces of a finite volume hyperbolic 3-manifold. We show that for arithmetic hyperbolic 3-manifolds that contain a totally geodesic surface, this spectrum determines the commensurability class. In addition, we show that any finite volume hyperbolic 3-manifold has many pairs of non-isometric finite covers with identical spectra. Forgetting multiplicities, we can also construct pairs where the volume ratio is unbounded.

Infinitely many meridional essential surfaces of bounded genus in hyperbolic knot exteriors

We show the existence of an infinite collection of hyperbolic knots where each of which has in its exterior meridional essential planar surfaces of arbitrarily large number of boundary components, or, equivalently, that each of these knots has essential tangle decompositions of arbitrarily large number of strings. Moreover, each of these knots has in its exterior meridional essential surfaces of any positive genus and (even) number of boundary components. That is, the compact surfaces that have a meridional essential embedding into a hyperbolic knot exterior have meridional essential embedding…

亏格2曲面上的柔性双曲锥度量

Flexible hyperbolic cone metrics on the genus 2 surface

本文研究亏格2曲面上负曲率双曲锥度量的刚性问题，借助 Erlandsson–Leininger–Sadanand 定理证明柔性度量等价类恰有九个映射类群轨道。

On quotient orbifolds of hyperbolic 3-manifolds of genus two

We analyze the orbifolds that can be obtained as quotients of hyperbolic 3-manifolds admitting a Heegaard splitting of genus two by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the hyperbolic 2-fold branched coverings of 3-bridge links. If the 3-bridge link is a knot, we prove that the underlying topological space of the quotient orbifold is either the 3-sphere or a lens space and we describe the combinatorial setting of the singular set for each possible isometry group. In the case of 3-bridge links with two or three components, the situation …

Asymmetric hyperbolic L-spaces, Heegaard genus, and Dehn filling

An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic geometry, Floer theory, and verified computer calculations. Of independent interest is our technique for using interval arithmetic to certify symmetry groups and non-existence of isometries of cusped hyperbolic 3-manifo…

Hyperbolic cone-manifold structures with prescribed holonomy II: higher genus

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of $2π$, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we build upon previous work with punctured tori to prove results for higher genus surfaces. Our techniques construct…

PL-genus of surfaces in homology balls

We consider manifold-knot pairs $(Y,K)$ where $Y$ is a homology sphere that bounds a homology ball. We show that the minimum genus of a PL surface $Σ$ in a homology ball $X$ such that $\partial (X, Σ) = (Y, K)$ can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from $(Y, K)$ to any knot in $S^3$ can be arbitrarily large. The proof relies on Heegaard Floer homology.

Free genus one knots with large volume

A Seifert surface F for a knot K is free if the complement of F is a handlebody (i.e., has free fundamental group). The free genus of K is the minimum genus among all free Seifert surfaces for K. In this paper we show that there exist families of hyperbolic knots with arbitrarily large volume, which each have free genus one. This implies that there are knots with free genus one but arbitrarily large canonical genus, and that there exist knots admitting incompressible free Seifert surfaces which cannot be built by applying Seifert's algorithm to a projection of the knot.

Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes (i.e guts) of the comp…

Isospectral hyperbolic surfaces of infinite genus

We show that any infinite-type surface without planar ends admits arbitrarily large families of length isospectral hyperbolic structures. If the surface has infinite genus and its space of ends is self-similar, we construct an uncountable family of isospectral and quasiconformally distinct hyperbolic structures.

三维球面中亏格为一的双曲纽结的Kakimizu复形

The Kakimizu complex for genus one hyperbolic knots in the 3-sphere

本文研究三维球面中亏格为一的双曲纽结的Kakimizu复形结构，证明其维数不超过4且由至多两个相交于公共面的单纯形组成，并构造了无穷多相应例子。

Embedding of metric graphs on hyperbolic surfaces

An embedding of a metric graph $(G, d)$ on a closed hyperbolic surface is \emph{essential}, if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus $g_e(G)$ of $(G, d)$ is the lowest genus of a surface on which such an embedding is possible. In the next result, we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ admits such an embedding (…

Minimal spheres of arbitrarily high Morse index

We construct a smooth Riemannian metric on any 3-manifold with the property that there are genus zero embedded minimal surfaces of arbitrarily high Morse index.

The simple separating systole for hyperbolic surfaces of large 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.

Heegaard genus and Property 'tau' for hyperbolic 3-manifolds

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $τ$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.

Rank and genus of 3-manifolds

We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also construct toroidal such examples containing hyperbolic JSJ pieces.

大亏格随机双曲面的紧长度谱

The tight length spectrum of large-genus random hyperbolic surfaces with many cusps

研究大亏格随机双曲面中紧测地线的长度统计，证明其在大亏格极限下恢复为泊松点过程，利用递推公式和推广的积分公式。

Generalized torsion for knots with arbitrarily high genus

In a group, a non-trivial element is called a generalized torsion element if some non-empty finite product of its conjugates equals to the identity. We say that a knot has generalized torsion if its knot group admits such an element. For a (2, 2q+1)-torus knot K, we demonstrate that there are infinitely many unknots c such that p-twisting K about c yields a twist family, which consists of hyperbolic knots with generalized torsion whenever |p| > 3. This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus…

Heegaard Splittings of Twisted Torus Knots

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain "twisted torus knots" originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. W…

On hyperbolic surface bundles over the circle as branched double covers of the $3$-sphere

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3…

Independence of volume and genus $g$ bridge numbers

A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is natural to extend this comparison to ask whether a $(g,b)$-bridge surface for a knot $K$ in $S^3$ carries any geometric information related to the knot exterior. In this paper, we show that (unlike in the case of Heegaard splittings) hyperbolic volume and genus $g$ bridge numbers are completely independent. That is, for any $g$, we construct explicit sequen…

Heegaard genera in congruence towers of hyperbolic 3-manifolds

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove that for any $ε>0$ there exist infinitely many congruence covers $\{M_i\}$ such that, for any $x \in M$, $M_i$ contains an embbeded ball $B_x$ (with center $x$) satisfying $\text{vol}(B_x) > (\text{vol}(M_i))^{\tfrac{1}{4}-ε}$. We get similar results in the arithmetic non-compact case.

Braids, entropies and fibered 2-fold branched covers of 3-manifolds

It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper concerns entropy of pseudo-Anosov monodromies for hyperbolic fibered $3$-manifolds. We prove that there exist infinitely many closed orientable $3$-manifolds $M$ such that the minimal entropy over all hyperbolic, genus $g$ surface bundles over the circle as $2$-fold branched covers of the $3$-manifold $M$ is comparable to $1/g$.

Large embedded balls and Heegaard genus in negative curvature

We show if M is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus g then g >= 1/2 cosh(r) where r denotes the radius of any isometrically embedded ball in M. Assuming an unpublished result of Pitts and Rubinstein improves this to g >= 1/2 cosh(r) + 1/2. We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls.

Bridge spectra of twisted torus knots

We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbolic knots exhibiting this property. In addition, we show that there are Berge and Dean knots with arbitrarily large genus one bridge numbers, and as a result, we give solutions to problems of Eudave-Muñoz concerning tunnel number one knots.

Length partition of random multicurves on large genus hyperbolic surfaces

We study the length statistics of the components of a random multicurve on a surface of genus $g \geq 2$. For each fixed genus, the existence of such statistics follows from the work of M.~Mirzakhani, F.~Arana-Herrera and M.~Liu. We prove that as the genus $g$ tends to infinity the statistics converge in law to the Poisson--Dirichlet distribution of parameter $θ=1/2$. In particular, as the genus tends to infinity the mean length of the three longest components converge respectively to $75.8\%$, $17.1\%$ and $4.9\%$ of the total length.

高亏格曲面精细曲线图的双曲等距

Hyperbolic isometries of the fine curve graph of higher genus surfaces

该研究针对闭双曲曲面上同痕于恒等映射的同胚，证明了其在精细曲线图上的双曲作用、相对于有限不变集同痕于pseudo-Anosov映射、遍历同调旋转集非空内部这三者的等价性。

Random hyperbolic surfaces of large genus have first eigenvalues greater than $\frac{3}{16}-ε$

Let $M_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $ε>0$, as genus $g$ goes to infinity, a generic surface $X\in M_g$ satisfies that the first eigenvalue $λ_1(X)>\frac{3}{16}-ε$. As an application, we also show that a generic surface $X\in M_g$ satisfies that the diameter $\mathrm{diam}(X)<(4+ε)\ln(g)$ for large genus.

Bridge number, Heegaard genus and non-integral Dehn surgery

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly irreducible, genus g Heegaard splitting of M then K intersects a thick level at most 2w(g) times. Typically, this shows that the bridge number of K with respect to this Heegaard splitting is at most w(g), and the tunnel number of K is at most w(g) + g-1.

双曲二 genus 表面分支结构的辛几何

On the symplectic geometry of branched hyperbolic surfaces in genus two

本文研究双曲二 genus 表面分支结构的辛几何，通过构造类似芬谢尔-尼尔斯坐标，证明其满足类似沃尔佩特魔公式，从而为戈登辛形式提供达布坐标。

Counting Essential Surfaces in a Closed Hyperbolic 3-Manifold

Let M be a closed hyperbolic 3-manifold. We show that the number of genus g surface subgroups of the fundamental group of M grows like g^{2g}.

高亏格典型双曲曲面上扭曲拉普拉斯算子的谱分布

Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus

本文研究高亏格典型双曲曲面上扭曲拉普拉斯算子的谱分布，通过调和形式的小上确界范数估计，证明了谱实部分布的一致Weyl定律。

亏格2的Teichmüller曲线的极小双曲面积

Minimal Hyperbolic Area of Teichmuller Curves in Genus Two

该研究确定亏格2的Teichmüller曲线的极小双曲面积为3π/5，明确其取得条件，通过结合轨形分类与仿射下降构造，排除非平方零点模式完成证明。

纽结弗洛尔同调中具有任意大挠率阶的双曲纽结

Hyperbolic knots with arbitrarily large torsion order in knot Floer homology

本文证明双曲纽结（主要是扭转环面纽结）可实现任意大的两种纽结弗洛尔同调挠率阶，并首次构造出共享同一 Upsilon 挠率函数的无限族双曲纽结。

Systoles of hyperbolic surfaces with big cyclic symmetry

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 $g\ge4$.

Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Let $X$ be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle $π$ is the identity. This space $X$ is closely related to the PSL$_2(\mathbb{R})$-character variety of the genus 2 surface $Σ$. In this article we study the topology and the natural symplectic structure on $X$, and we describe the action of the mapping class group of $Σ$ on $X$. This completes the classification of the ergodic components of the character variety in genus 2 initiated in our previous work.

大亏格随机双曲曲面上的非简单收缩

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$\log g$。

Dehn surgeries on knots which yield lens spaces and genera of knots

Let $K$ be a hyperbolic knot in the 3-sphere. If $r$-surgery on $K$ yields a lens space, then we show that the order of the fundamental group of the lens space is at most $12g-7$, where $g$ is the genus of $K$. If we specialize to genus one case, it will be proved that no lens space can be obtained from genus one, hyperbolic knots by Dehn surgery. Therefore, together with known facts, we have that a genus one knot $K$ admits Dehn surgery yielding a lens space if and only if $K$ is the trefoil.

The concordance genus of knots

In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <= g_c(K) <= g(K). Casson and Nakanishi gave examples to show that g_4(K) need not equal g_c(K). We begin by reviewing and extending their results. For knots representing elements in A, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon's result that A is nontrivial implies that g_4(K) ca…

Asymptotic dimension and the disk graph II

We show that the asymptotic dimension of a hyperbolic relatively hyperbolic graph is finite provided that this holds true uniformly for the peripheral subgraphs and for the electrifiation. We use this to show that the asymptotic dimension of the disk graph of a handlebody of genus at least two is at most quadratic in the genus.

Parabolic generating pairs of genus-one 2-bridge knot groups

We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups to genus-one hyperbolic 2-bridge knot groups.

小亏格纽结上的透镜空间手术

Lens space surgeries on knots of small genus

本文证明了亏格至多5的纽结中，仅 $(-2,3,7)$ 扭结有透镜空间手术，并通过关键技术结果结合先前工作验证了 Berge 猜想。

Thick surfaces in hyperbolic 3-manifolds

We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.

Knots with unique minimal genus Seifert surface and depth of knots

We describe a procedure for creating infinite families of hyperbolic knots having unique minimal genus Seifert surface. A large subset of these knots have the further property that the surface cannot be the sole compact leaf of a depth one foliation of the knot exterior.

Non-isometric hyperbolic 3-orbifolds with the same topological type and volume

We construct pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume. Topologically these orbifolds are mapping tori of pseudo-Anosov maps of the surface of genus 2, with singular locus a fibred (hyperbolic) link with five components.

Mirzakhani的亏格较大且具有许多尖点的双曲曲面上的简单闭测地线频率

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。

构造一个具有任意小特征值的闭合双曲面

Construction of a Closed Hyperbolic Surface of Arbitrarily Small Eigenvalue of Prescribed Serial Number

本文通过几何方法构造了具有任意小第n个特征值的闭合双曲面，改进了Randol 1974年的结果。

Finite-volume hyperbolic 4-manifolds that share a fundamental polyhedron

It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by constructing a sequence of finite-sided finite-volume polyhedra with side-pairings that yield manifolds. In fact, we show that arbitrarily many nonhomeomorphic hyperbolic 4-manifolds may share a fundamental polyhedron. As a by-product of our examples, we also show in a constructive way that the set of volumes of hyperbolic 4-manifolds contains the set of even in…

Geometrization of almost extremal representations in $\text{PSL}_2\Bbb R$

Let $S$ be a closed surface of genus $g$. In this paper, we investigate the relationship between hyperbolic cone-structure on $S$ and representations of the fundamental group into $\text{PSL}_2\Bbb R$. We consider surfaces of genus greater than $g$ and we show that, under suitable conditions, every representation $ρ:π_1 S\longrightarrow \text{PSL}_2\Bbb R$ with Euler number $\mathcal{E}(ρ)=\pm\big(χ(S)+1\big)$ arises as holonomy of a hyperbolic cone-structure $σ$ on $S$ with a single cone point of angle $4π$. From this result, we derive that for surfaces of genus $2$ every representation with …

超薄结的Kakimizu复形的线性上界

A Linear Bound on the Diameter of the Kakimizu Complex for Hyperbolic Knots

本文通过研究超薄结的Kakimizu复形，证明其直径随曲面 genus 线性增长，并给出上界为$6g-4$。

Mapping classes associated to mixed-sign Coxeter graphs

We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of symmetries. The periodic mapping classes can in turn be used to construct sequences of pseudo-Anosov mapping classes with bounded normalized dilatation and arbitrarily high genus. We show that the smallest known accumulation point of normalized dilatations can be realized by such a sequence.

Unicellular maps vs hyperbolic surfaces in large genus: simple closed curves

We study uniformly random maps with a single face, genus $g$, and size $n$, as $n,g\rightarrow \infty$ with $g = o(n)$, in continuation of several previous works on the geometric properties of "high genus maps". We calculate the number of short simple cycles, and we show convergence of their lengths (after a well-chosen rescaling of the graph distance) to a Poisson process, which happens to be exactly the same as the limit law obtained by Mirzakhani and Petri (2019) when they studied simple closed geodesics on random hyperbolic surfaces under the Weil-Petersson measure as $g\rightarrow \infty$…

大亏格随机双曲曲面上同调类中的闭测地线

Closed geodesics in homology classes on random hyperbolic surfaces of large genus

研究大亏格随机双曲曲面上同调类中闭测地线分布，通过将曲面视为模空间随机点，研究闭测地线加权计数函数波动，得出大亏格极限下不同模的方差情况，与胡利猜想对比并给出差异解释。

Bounding canonical genus bounds volume

A Seifert surface for a knot K is called canonical if it can be built by applying Seifert's algorithm to some projection of K. The canonical genus of K is the smallest genus of a surface so obtained. In this paper we show that there is a bound on the volume of a hyperbolic knot which admits a canonical surface of genus g. The bound can, in fact, be chosen to be linear in g.