Curves on surfaces, charts, and words

Simple closed curves, word length, and nilpotent quotients of free groups

We consider the fundamental group $π$ of a surface of finite type equipped with the infinite generating set consisting of all simple closed curves. We show that every nilpotent quotient of $π$ has finite diameter with respect to the word metric given by this set. This is in contrast with a result of Danny Calegari that shows that $π$ has infinite diameter with respect to this set. Furthermore, we give a general criterion for a finitely generated group equipped with a generating set to have this property.

A remark on the word length in surface groups

Let $Σ$ be a surface of negative Euler characteristic and $S$ a generating set for $π_1(Σ,p)$ consisting of simple loops that are pairwise disjoint (except at $p$). We show that the word length with respect to $S$ of an element of $π_1(Σ,p)$ is given by its intersection number with a well-chosen collection of curves and arcs on $Σ$. The same holds for the word length of (a free homotopy class of) an immersed curve on $Σ$. As a consequence, we obtain the asymptotic growth of the number of immersed curves of bounded word length, as the length grows, in each mapping class group orbit.

从曲线到文字再回到曲线：最小面积同伦的几何计算

From Curves to Words and Back Again: Geometric Computation of Minimum-Area Homotopy

本文统一了Blank与Nie的组合文字方法，给出Nie算法正确性的构造性几何证明，并提出首个计算闭曲线最小面积自重叠分解的多项式时间算法。

Partitions of cyclic words and Goldman-Turaev Lie bialgebra

The free $\mathbb{Z}$-module generated from the set of non-trivial homotopy classes of closed curves on an oriented surface has the structure of Lie bialgebra by two operations, the Goldman bracket and Turaev cobracket. M. Chas gave a combinatorial redefinition of these two operations through a natural identification of the homotopy classes of closed curves on the surface with the cyclic words in the generators and their inverses of the fundamental group of the surface. We present a new approach to give a combinatorial definition of the bracket and cobracket, focusing on the information given …

$2$-stratifold groups have solvable Word Problem

$2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed curves where several sheets meet. We show that the word problem for fundamental groups of $2$-stratifolds is solvable.

Algorithm for filling curves on surfaces

Let $Σ$ be a compact, orientable surface of negative Euler characteristic, and let $h$ be a complete hyperbolic metric on $Σ$. A geodesic curve $γ$ in $Σ$ is filling, if it cuts the surface into topological disks and annuli. We propose an efficient algorithm for deciding whether a geodesic curve, represented as a word in some generators of $π_1(Σ)$, is filling. In the process, we find an explicit bound for the combinatorial length of a curve given by its Dehn-Thurston coordinate, in terms of the hyperbolic length. This gives us an efficient method for producing a collection which is guaranteed…

来自Yang-Baxter算子的曲面辫子不变量及其上同调

Surface Braid Invariants from Yang-Baxter Operators and Their Cohomology

本文提出利用Yang-Baxter算子通过辫子系统构造曲面辫子不变量，并借助图表使用其3-上循环，推广了quandle上循环不变量，讨论了闭包定义纽结曲面不变量的方法。

Virtual Strings for Closed Curves with Multiple Components

A Gauss paragraph is a combinatorial formulation of a generic closed curve with multiple components on some surface. A virtual string is a collection of circles with arrows that represent the crossings of such a curve. Every closed curve has an underlying virtual string and every virtual string has an underlying Gauss paragraph. A word-wise partition is a partition of the alphabet set of a Gauss paragraph that satisfies certain conditions with respect to the Gauss paragraph. In this paper we use the theory of virtual strings to obtain a combinatorial description of closed curves in the 2-spher…

Combinatorial Lie bialgebras of curves on surfaces

Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a generalization of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we fo…

Combinatorialization of spaces of nondegenerate spherical curves

A parametric curve $γ$ of class $C^n$ on the $n$-sphere is said to be nondegenerate (or locally convex) when $\det\left(γ(t),γ'(t),\cdots,γ^{(n)}(t)\right)>0$ for all values of the parameter $t$. We orthogonalize this ordered basis to obtain the Frenet frame $\mathfrak{F}_γ$ of $γ$ assuming values in the orthogonal group $\operatorname{SO}_{n+1}$ (or its universal double cover, $\operatorname{Spin}_{n+1}$), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve $γ$ we assign its itinerary: a word $w$ in the alphabet $S_{n+1}\smallsetminus\{e\}$ that encodes the successio…

The word problem for 3-manifolds built from injective handlebodies

This paper gives a proof that the fundamental group of a class of closed orientable 3-manifolds constructed from three injective handlebodies has a solvable word problem. This is done by giving an algorithm to decide if a closed curve in the manifold is null-homotopic. Non-Haken and non-Seifert fibered examples are constructed by performing Dehn surgery on a class of two-bridge knots.

具有谢尔宾斯基曲线边界和门格曲线边界的双曲考克斯特群的完全刻画

Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary

本文用神经给出了格罗莫夫边界为谢尔宾斯基曲线或门格曲线的字双曲考克斯特群的完全刻画，主要基于对已有文献结果的组合。

Experiments suggesting that the distribution of the hyperbolic length of closed geodesics sampling by word length is Gaussian

Each free homotopy class of directed closed curves on a surface with boundary can be described by a cyclic reduced word in the generators of the fundamental group and their inverses. The word length is the number of letters of the cyclic word. If the surface has a hyperbolic metric with geodesic boundary, the geometric length of the class is the length of the unique geodesic. By computer experiments, we investigate the distribution of the geometric length among all classes with a given word length in the pair of pants surface. Our experiments strongly suggest that the distribution is normal.

Word hyperbolic extensions of surface groups

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves is a quasi-isometric embedding.This in turn is equivalent to G/H being convex cocompact in the sense of Farb and Mosher.

On the combinatorics of exact Lagrangian surfaces

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated to produce more such skeleta --- in many cases, infinitely many more. Each skeleton is built around a Lagrangian surface. Passing to the Fukaya category, the skeletal surgeries induce cluster transformations on the spaces of rank one local systems on these surfaces, and noncommut…

Measuring complexity of curves on surfaces

We consider the relations between different measures of complexity for free homotopy classes of curves on a surface $Σ$, including the minimum number of self-intersections, the minimum length of the words representing them in a geometric presentation of $π_1(Σ)$, and the minimum degree of the coverings of $Σ$ to which they lift as embeddings.

Finite-type invariants for curves on surfaces

This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-order invariants for plane immersed curves. The invariants in this theory are developed using the word theory proposed by Turaev.

Counting intersections of normal curves

A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the composition by an exotic matrix multiplication. An efficient solution of the word problem for mapping class groups of punctured surfaces is proposed, with efficiency understood in a more restrictive way than the most common one.

局部凸曲线空间的路径分层

Stratification of spaces of locally convex curves by itineraries

本文通过定义局部凸曲线的路径，将曲线空间按路径分层为可缩子流形，证明相关猜想，并为构建CW复形提供基础。

关于环面曲线图上 Anosov 映射的平移长度

On Translation Lengths of Anosov Maps on Curve Graph of Torus

本文证明环面曲线图上的 Anosov 映射具有测地轴且其稳定平移长度为正整数，并给出计算精确平移长度的算法，应用于确定最小平移长度字、建立平移长度比的有效界限及估计 Anosov 映射数量的整体增长。

曼哈顿曲线、拓扑流的遍历理论与刚性

The Manhattan curve, ergodic theory of topological flows and rigidity

本文为一般非初等双曲群引入曼哈顿曲线，利用拓扑流遍历理论和Patterson-Sullivan测度证明其可微性与刚性，并给出三角形群等显式计算。