The space of Kleinian punctured torus groups is not locally connected

Deformation spaces of Kleinian surface groups are not locally connected

For any closed surface $S$ of genus $g \geq 2$, we show that the deformation space of marked hyperbolic 3-manifolds homotopy equivalent to $S$, $AH(S \times I)$, is not locally connected. This proves a conjecture of Bromberg who recently proved that the space of Kleinian punctured torus groups is not locally connected. Playing an essential role in our proof is a new version of the filling theorem that is based on the theory of cone-manifold deformations developed by Hodgson, Kerckhoff, and Bromberg.

Number of Systoles of Once-Punctured Torus and Four-Punctured Sphere

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.

环面计数与Kleinian群的自连接

Torus counting and self-joinings of Kleinian groups

该论文针对Kleinian群自连接下的一类d维环面堆积，建立了小体积环面的计数与等分布渐近公式，推广了Oh-Shah的圆堆积结果。

SL(2,R)-表示的谱：一次 punctured torus 情况

Spectrum of SL(2,R)-characters: the once-punctured torus case

研究SL(2,R)表示的谱，特别是一次 punctured torus 的情况，证明通用表示的谱是康托尔集，并描述其与区间交换变换相关局部常数上同调的动力学。

Dynnikov coordinates on punctured torus

We generalize Dynnikov coordinate system previosly defined on the standard punctured disk to an orientable surface of genus-1 with n punctures and one boundary component.

Adjoint Reidemeister torsions of once-punctured torus bundles

Gang, Kim and Yoon have recently proposed a conjecture on a vanishing identity of adjoint Reidemeister torsions of hyperbolic 3-manifolds with torus boundary, from the viewpoint of wrapped M5-branes. In this paper, we provide infinitely many new supporting examples to this conjecture. These examples come from hyperbolic once-punctured torus bundles. We show that the vanishing identity holds for all hyperbolic once-punctured torus bundles with tunnel number one. We also show the vanishing identity does not hold for any torus knot exteriors.

Guts of surfaces in punctured-torus bundles

Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic JSJ-subpair of M-S and show, in particular, that the guts of (M,S) is empty.

On canonical triangulations of once-punctured torus bundles and two-bridge link complements

We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle.

2-穿孔环面与5-穿孔球面上伪Anosov映射的Agol环

Agol cycles of pseudo-Anosov maps on the 2-punctured torus and 5-punctured sphere

本文研究了2-穿孔环面和5-穿孔球面上一族伪Anosov映射的Agol环，给出了测度轨道并计算其Agol环，提出了映射共轭的条件，并发现了伸缩因子的新公式。

Generalized torsions in once punctured torus bundles

A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without generalized torsions, it is conjectured that 3-manifold groups without generalized torsions are bi-orderable. In this paper, we find generalized torsions in the fundamental groups of once punctured torus bundles which are not bi-orderable. Our result contains a generalized torsion in a tunnel number two hyperbolic once punctured torus bundle.

Branched Cauchy-Riemann Structures on Once-Punctured Torus Bundles

Unlike in hyperbolic geometry, the monodromy ideal triangulation of a hyperbolic once-punctured torus bundle $M_f$ has no natural geometric realisation in Cauchy-Riemann (CR) space. By introducing a new type of $3$--cell, we construct a different cell decomposition $\mathcal{D}_f$ of $M_f$ that is always realisable in CR space. As a consequence, we show that every hyperbolic once-punctured torus bundle admits a branched CR structure, whose branch locus is the set of edges of $\mathcal{D}_f$. Furthermore, we explicitly compute the ramification order around each component of the branch locus and…

Thurston's bending measure conjecture for once punctured torus groups

We prove Thurston's bending measure conjecture for quasifuchsian once punctured torus groups. The conjecture states that the bending measures of the two components of the convex hull boundary uniquely determine the group.

关于全穿孔伪阿诺索夫映射的归一化伸缩系数的集合

On the set of normalized dilatations of fully-punctured pseudo-Anosov maps

本文改进全穿孔伪阿诺索夫映射环面 veering 三角剖分中四面体数量的上界，并结合计算方法确定归一化伸缩系数集合的最小元素为黄金比例的平方、最小聚点为其四次方。

Characterizing right-veering homeomorphisms of the punctured torus via the Burau representation

We classify right-veering homeomorphisms of the once-punctured torus using the Burau representation of the 3-strand braid group. We show that reducible and periodic mapping classes in B_3 can be identified as right-veering by consideration of the reduced version of the Burau representation. Given any element beta in B_3, we give a method to quickly determine its action on the generators of the fundamental group of the 3-times punctured disk. This action which determines whether beta is right-veering, left-veering, or neither.

The canonical decomposition of once-punctured torus bundles

In this paper, we determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defined by Floyd and Hatcher. Unlike previous work on this problem, the techniques we use are not geometric. Instead, they involve angled polyhedral decompositions, thin position and a version of almost normal surface theory.

Kleinian groups and the complex of curves

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have `bounded geometry' (lower bounds on injectivity radius) in terms of a sequence of coefficients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of punctured-torus surface groups. In that setting the ending invari…

Combinatorial $k$-systoles on a punctured torus and a pair of pants

In this paper, $S$ denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of \emph{\textbf{combinatorial $k$-systoles}} that is closed curves with self-intersection numbers greater than $k$ and with least combinatorial length. We show that the maximal intersection number $I^c_k$ of combinatorial $k$-systoles of $S$ grows like $k$ and $\underset{k\rightarrow+\infty}{\limsup}(I^c_k-k)=+\infty$. This result, in case of a pair of pants and a punctured torus, is a positive response to the combinatorial version of the Erlandsson - Parlier conjecture, origin…

Nonorientable, incompressible surfaces in punctured-torus bundles over S^1

We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to the projection of the bundle into the basis S^1.

关于一次穿孔环面群的凸核体积的渐近性

On the Asymptotics of Convex Core Volumes of Once-Punctured Torus Groups

研究一次穿孔环面上与伪阿诺索夫映射类$\psi$相关的拟富克斯流形序列的凸核体积渐近行为，证明其凸核体积与$\psi$映射环面体积的$2n$倍相差至多一个有界常数。

The Tri-Pants Graph of the Twice-Punctured Torus

We investigate the structure of the tri-pants graph, a simplicial graph introduced by Maloni and Palesi, whose vertices correspond to particular collections of homotopy classes of simple closed curves of the twice-punctured torus, called tri-pants, and whose edges connect two vertices whenever the corresponding pants differ by suitable elementary moves. In particular, by examining the relationship between the tri-pants graph and the dual of the Farey complex, we prove that the tri-pants graph is connected and has infinite diameter.

Triangulated cores of punctured-torus groups

We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an Epstein-Penner convex hull construction in Minkowski space. The result extends to certain non-quasifuchsian punctured-torus groups, and in fact to all of them if a strong version of the Pleating Lamination Conjecture is true.

Tessellating the moduli space of strictly convex projective structures on the once-punctured torus

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of framed strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.

Cusp excursions for the earthquake flow on the once-punctured torus

In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow and the horocyclic flow behave very differently in cusp excursions. In particular, we prove a relation between the systole function and continued fractions and discuss the cusp excursions of earthquake trajectories.

(1,1)-knots via the mapping class group of the twice punctured torus

We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG(T,2) with a standard element only depending on the ambient space. As a notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k,ck+2).

单孔环面上的单自交单词及其计数

Single self-intersection words on the once-punctured torus and their counting

研究单孔环面上单自交闭曲线，从组合角度分类，借此确定给定单词长度且自交数为零、一及任意数的闭曲线数量。

A new identity for SL(2,C)-characters of the once punctured torus group

We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.

Earthquakes on the once-punctured torus

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space to examine earthquake deformations further. Two families of curves are used as examples. E…

带一个孔的环面上给定字长的精确曲线计数

Exact curve counting of given word length on the once-punctured torus

针对带一个孔的环面，研究人员得出映射类群轨道内给定字长曲线数的精确公式，解决了Cha在2016年提出的相关猜想。

Virtually Haken surgeries on once-punctured torus bundles

We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of ``most'' commensurability classes of punctured torus bundles. In particular, we define an integer-valued complexity function on monodromies $f$ (essentially the length of the LR-factorization of $f_*$ in $PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$ has co…

Coordinates for quasi-Fuchsian punctured torus space

We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely the classical Fuchsian space with Fenchel-Nielsen coordinates and the Maskit embedding. We also show how they relate to the pleating coordinates of Keen and Series.

Pleating coordinates for the Teichmüller space of a punctured torus

We construct new coordinates for the Teichmüller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_μ$ that depend holomorphically on a parameter $μ$ varying in a simply connected domain in $\bold{C}$. They describe the geometry of the hyperbolic manifold $\bold{H}^3/G_μ$; they reflect exactly the visual patterns one sees in the limit sets of the groups $G_μ$; and they are directly computable from the generators of $G_μ$.

Cannon-Thurston Maps,i-bounded Geometry and a theorem of McMullen

The notion of i-bounded geometry generalises simultaneously bounded geometry and the geometry of punctured torus Kleinian groups. We show that the limit set of a surface Kleinian group of i-bounded geometry is locally connected by constructing a natural Cannon-Thurston map. This is an exposition of a special case of the main result of arXiv:math/0607509.

Detection of incompressible surfaces in hyperbolic punctured torus bundles

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We convert non fiber incompressible surfaces from their form to the form output by Yoshida's construction, and run his construction backwards to give (for non semi-fibers, which we identify) the data needed to construct ideal points of the deformation variety corresponding to those surfac…

通过扭曲Alexander多项式的单穿孔环面丛的射影刚性

Projective Rigidity of Once-Punctured Torus Bundles via the Twisted Alexander Polynomial

本文通过扭曲Alexander多项式验证双曲单穿孔环面丛的射影刚性，并将其与字符簇切空间上的诱导作用及谱序列中的群论作用联系起来。

A characteristics for a surface sum of two handlebodies along an annulus or a once-punctured torus to be a handlebody

The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum $H=H_1\cup_A H_2$ of two handlebodies $H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a longitude for either $H_1$ or $H_2$. 2. Let $H=H_1\cup_T H_2$ be a surface sum of two handlebodies $H_1$ and $H_2$ along a once-punctured torus $T$. Suppose that $T$ is incompressible in both $H_1$ and $H_2$. Then $H$ is a handlebody if and only if the there exists a collection $\{δ, σ\}$ of simple clos…

Cluster Algebra and Complex Volume of Once-Punctured Torus Bundles and Two-Bridge Knots

We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.

Character varieties of once-punctured torus bundles with tunnel number one

We determine the PSL_2(C) and SL_2(C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine `natural' models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Ad…

穿孔环面丛上的非标准双向序

Non-standard bi-orders on punctured torus bundles

本文研究穿孔环面丛群的非标准双向序，证明其最大及次大真凸子群与标准双向序一致，并确认这些群确实存在非标准双向序。

Azumaya algebras and once-punctured torus bundles

Recent work of Chinburg, Reid, and Stover has shown that certain arithmetic and algebro-geometric properties of the character variety of a hyperbolic knot complement in the 3-sphere $M=S^3\setminus K$ yields topological and number theoretic information about Dehn fillings of M. Specifically, they show how the study of a certain extension problem for quaternion Azumaya algebras is related to topological invariants associated to these fillings. In this paper, we extend their work to the setting of hyperbolic once punctured torus bundles. Along the way, we exhibit new phenomenon in the relevant e…

On hyperbolic once-punctured-torus bundles III: Comparing two tessellations of the complex plane

To each once-punctured-torus bundle, $T_ϕ$, over the circle with pseudo-Anosov monodromy $ϕ$, there are associated two tessellations of the complex plane: one, $Δ(ϕ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the other, $CW(ϕ)$, is a fractal tessellation given by the Cannon-Thurston map of the fiber group switching back and forth between gray and white each time it passes through $\infty$. In this paper, we study the relation between $Δ(ϕ)$ and $CW(ϕ)$.

4穿孔球面与1穿孔环面的skein代数的强正性

Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus

本文借助复三次曲面的高亏格Gromov–Witten镜像对称构造刻画4穿孔球面skein代数，并证明其与1穿孔环面bracelets基结构常数的强正性。

Irreducible triangulations of the once-punctured torus

A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the once-punctured torus, consisting of exactly 297 non-isomorphic triangulations.

Fourier transform for quantum $D$-modules via the punctured torus mapping class group

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attached to $H$. We show that the elliptic double is the universal source of such representations. We recover the representations of the punctured torus braid group obtained in arXiv:0805.2766, and hence construct a homomorphism to the Heisenberg double $D_H$, which is an isomorphism if…

Examples of relative deformation spaces that are not locally connected

We provide an infinite family of pared manifolds whose relative deformation spaces of hyperbolic structures on these manifolds are not locally connected. This is a natural extension of the recent result of Bromberg that shows the space of Kleinian punctured torus groups is not locally connected.

关于一次 punctured torus 系列的特征品种拓扑

On the topology of character varieties of once-punctured torus bundles

本文研究了一次 punctured torus 系列的特征品种拓扑，通过分析纤维上的特征并构建分支覆盖映射，扩展了 Boyer 等人的成果，并展示了无限家族的超几何 bundles 的 canonical curves 特征具有无界 genus。

Pleating invariants for punctured torus groups

In this paper we give a complete description of the space $ \QF $ of quasifuchsian punctured torus groups in terms of what we call {\em pleating invariants}. These are natural invariants of the boundary $\bch$ of the convex core of the associated hyperbolic 3-manifold $M$ and give coordinates for the non-Fuchsian groups $\QF - \F$. The pleating invariants of a component of $\bch$ consist of the projective class of its bending measure, together with the lamination length of a fixed choice of transverse measure in this class. Our description complements that of Minsky in \cite{MinskyPT}, in whic…

The Maskit embedding of the twice punctured torus

The Maskit embedding M of a surface Σis the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom' end consists of two triply punctured spheres, the remains of Σwhen two fixed disjoint curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichmüller space T(Σ). We investigate M when Σis a twice punctured torus, using the method of pleating rays. Fix a projective measure class [μ] supported on closed curves on Σ. The pleating r…

Self-intersection numbers of curves on the punctured torus

The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the generators of the fundamental group and their inverses: it is less than or equal to (L-2)^2/4 if L is even, and (L-1)(L-3)/4 if L is odd. The classes attaining this bound are explicitly described in terms of the generators; there are (L-2)^2 + 4 of them if L is even, and 2(L-1)(L-3) + 8 if L is odd; similar descriptions and totals are given for classes with self-i…

Khovanov homology and the Fukaya category of the traceless character variety for the twice-punctured torus

We describe a strategy for constructing reduced Khovanov homology for links in lens spaces by generalizing a symplectic interpretation of reduced Khovanov homology for links in $S^3$ due to Hedden, Herald, Hogancamp, and Kirk. The strategy relies on a partly conjectural description of the Fukaya category of the traceless $SU(2)$ character variety of the 2-torus with two punctures. From a diagram of a 1-tangle in a solid torus, we construct a corresponding object $(X,δ)$ in the $A_\infty$ category of twisted complexes over this Fukaya category. The homotopy type of $(X,δ)$ is an isotopy invaria…

小曲面的$\rm{PGL}_2$ skein代数

$\rm{PGL}_2$ skein algebras of small surfaces

本文研究小曲面（闭环面与一次穿孔环面）的$\rm{PGL}_2$ skein代数，给出显式有限表示、显式基及其乘法结构，并计算自然映射的核与代数中心，结果依赖于量子参数$q$。

Once-punctured tori and knots in lens spaces

We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.

一次穿孔环面上的$K$-理论Coulomb分支的对偶性

Dualities of $K$-theoretic Coulomb branches from a once-punctured torus

本文研究一次穿孔环面的量子化特征簇，发现三个不变子代数同构于量子化K-理论Coulomb分支，并受映射类群置换，证实了4维N=2*理论对偶性的物理预测。

关于双穿孔环面 skein 代数的双仿射 Hecke 代数的一个注记

A note on double affine Hecke algebra for skein algebra on twice-punctured torus

本文通过 Iwahori--Hecke 算子的 Heegaard 对偶，为双穿孔环面上的 skein 代数构造了 $C^\vee C_1$ 型双仿射 Hecke 代数的推广，并证明其自同构对应 Dehn 扭转，同时给出经典极限的簇代数构造。

一次 punctured torus 中 η 修正项的闭式公式

Closed Formulas for $η$-Corrections in the Once Punctured Torus

研究一次 punctured torus 的 Kauffman bracket skein 代数中 η 修正项，给出由原始行列式为二的配生成的 Chebyshev-threaded 家族的闭式公式，以及原始最大-thread 情况下的 η 线性 cascade 公式。