# Questions tagged [teichmuller-theory]

The teichmuller-theory tag has no usage guidance.

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### Modulus of image of a curve family in a rectangle

I don't expect to get a positive answer to this question but I may as well try.
Let $R$ be the rectangle in $\mathbb{C}$ given by $\{z=x+iy: 0\leq x \leq l, 0 \leq y \leq h\}$ for some $l,h>0$. ...

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### Extremal metric for image of a curve family

Let $U\subset \mathbb{C}$ be a domain and $\Gamma$ some family of curves in $U$ with $\textrm{mod}(\Gamma)<\infty$ and such that $\rho$ is an extremal metric for the modulus. Suppose we are given a ...

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### a normal subgroup of a triangle group

Let G = $<a,b : a^2= b^n = 1 >$ be the (2,n,$\infty$)-triangle group. Define a map $\sigma:G \to Z_2 \times Z_n$ via $a \mapsto (-1,1), b \mapsto (1,[1]).$ The kernel H of $\sigma$ is then a ...

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### Length functions on Teichmuller space with constant difference

Let $S$ be a closed oriented surface of genus $g\geq 2$. Let $\mathcal{T}$ be the corresponding Teichmuller space. Given a free homotopy class of closed curve $[\gamma]$ we can define the length ...

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### Reference request for quantum Teichmuller space

I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill ...

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### How was the pair of pants introduced [closed]

There are many results mentioned pairs of pants, and it seems to be a classical model. Why are the pairs of pants so useful?
For example, does it have any application if we estimate the perimeter or ...

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### What is a geodesic in Outer space?

The Culler-Vogtmann Outer space $\text{CV}_n$ is an analogue of Teichmuller space for the group $\text{Out}(F_n)$.
Is there any notion of a geodesic path in $\text{CV}_n$? Are there different ...

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### Geodesic current supported on a pencil?

Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...

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### Teichmueller disk and the $\mathrm{SL}_2\mathbb{R}$ action

Let $(X,\omega)$ be a Riemann surface of genus $g$ with holomorphic 1-form $\omega$ (or equivalently a translation structure). Let $\Omega\mathcal{T}_g$ be the space of holomorphic 1-forms over genus $...

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### Compactifications of SL(2)-character varieties of surfaces

Thurston compactified the Teichmuller space ${\cal T}(F)$ of a closed, oriented surface $F$ with a piecewise-linear sphere. Furthermore, as far as I understand, its linear pieces have natural ...

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### Classifying transverse curves to a surface foliation carried by a train track

Suppose that a foliation $\cal F$ on a surface $F$ is carried by a train track $\tau$. Is it possible to classify all $\cal F$-transverse multi-loops in $F$ in terms of a combinatorial data on $\tau$ (...

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### Parametrisation of Teichmüller space in terms of harmonic Beltrami differentials

I'm trying to learn Teichmüller theory, but appear to get stuck early on. Let $\Sigma$ be a smooth closed oriented surface of genus $g\geqslant 2$ and let $\mathrm{Conf}(\Sigma)$ denote the set of ...

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### Hyperelliptic locus is a $K(\pi,1)$

It is said in many papers that the hyperelliptic locus $\mathcal{H}_g\subseteq \mathcal{M}_g$ is a $K(\pi,1)$. (in the sense of orbifolds). This is justified by saying that it can be constructed as an ...

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### Nielsen-Thurston decomposition from the product of Dehn twists

Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is
Given an element as a product of Dehn twist, is it possible ...

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### Teichmuller uniqueness theorem with marked points

Let $S$ be a genus $g$, $g > 1$ Riemann surface, and let $h \colon S \to S$ be a homeomorphism of $S$. We denote by $[h] \in \text{Map}(S)$ the corresponding element of the mapping class group of $...

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### Real section of moduli space of Riemann surfaces

In (https://www.sciencedirect.com/science/article/pii/002240499390049Y) it is mentioned the real section of the moduli space of Riemann surfaces of genus 0. It can be intuitively defined as a subset ...

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### Can every curve be made transversal to a foliation by applying a pseudo-Anosov?

Let $F$ be a compact oriented surface with a foliation $\cal F$ with $k$-prong singularities only (or, if it helps, assume that $\cal F$ admits an invariant measure). Is it true then there exists a ...

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### Powers of pseudo-Anosov and the geometric intersection numbers

Let $\phi$ be a pseudo-Anosov of a compact oriented surface $F$ with boundary. Let $\beta\subset F$ be a simple closed loop and $\alpha$ either a simple closed loop or an embedded arc with endpoints ...

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### Putting a transverse measure on a surface foliation

Let $F$ be an orientable surface with a foliation $\cal F$ with $k$-prong singularities only, for $k\geq 3$.
Since I am looking for an invariant transverse measure on $\cal F$, assume that there is ...

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### Are isotopic transversal curves on a foliated surface transversally isotopic?

Let $F$ be an orientable surface (possibly with boundary) with a foliation $\cal F$ with $k$-prong saddle singularities only for $k\geq 3,$ (as in figure borrowed from Farb-Margalit book). Suppose ...

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### Confusion about Teichmuller curves and $SL_2$ action

Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $...

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### Ending lamination theorem

Let $M$ a compact manifold with surfaces $S_1,...,S_p$ as boundaries. Let us suppose that $M$ admits a complete hyperbolic structure. Then, from the ending lamination theorem, given either laminations ...

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### Length of a simple closed curve under Pseudo-Anosov maps

Let $S$ be a fixed hyperbolic surface with genus $g$ and $n$ punctures. Given any pseudo-Anosov map $f$ on $S$ (with stretch factor $\lambda$) with stable and unstable measured foliations $\mu^s$ and $...

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### Are pseudo-Anosov foliations dense?

A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of ...

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### Homological criterion for existence of a square root of a quadratic differential

Let $X$ be a compact Riemann surface, and let $q \in K^{\otimes2}(X)$ be a holomorphic quadratic differential
on $X$. Let $\Lambda_{q}$ be the sheaf of holomorphic vector fields $\chi$ satisfying
$\...

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### Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface

If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups:
$$\text{Mod}(\mathcal{R}')\longrightarrow \...

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### Shrinking the boundary of a Riemann surface

Let $X$ be a compact Riemann surface with boundary. Let us shrink each connected component of the boundary into a point. We get a closed topological surface $Z$ with several marked points (which came ...

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### Teichmuller groupoids in Grothendieck's esquisse d'un programme

Grothendieck in his Esquisse d'un programme mentioned without any precise definition and construction that the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the whole "tower" ...

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### Universal covering of a 2-sphere without $n$ points

Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic ...

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### Hyperbolic Metric on a Riemann Surface

From uniformization theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ punctures such that $2g+n\ge 3$ contains a unique hyperbolic metric. The ...

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### Degenerate Beltrami equation and inverse

The Beltrami equation $f_{\bar{z}}=\mu(z)f_{z}$ is degenerate when $\left \| \mu \right \|_{\infty}=1$. For these equations, Lehto and David among others have given conditions for existence. The Lehto ...

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### About the boundary of a fibered cone of a 3-manifold

Let $S$ be a closed surface, $\psi$ a pseudo-Anosov map, $M$ be the mapping torus, $\tilde{S}$ be a $\mathbb{Z}$-fold cover of $S$ using an invariant cohomology class. Let $D$ be a fundamental domain, ...

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### Relation between point pushing pseudo-Anosov map and the minimum length

Let $S$ be a closed hyperbolic surface. Suppose $Mod(S)$ denotes the mapping class groups and $T(F)$ denotes the Teichmüller space.
By Birman exact sequence we get the point pushing map $Push:\pi_1(S,...

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### Parametrizations of the Moduli Space of Riemann Surfaces

I am looking for a reference or references about different parameterizations of moduli space of Riemann surfaces of genus $g$ with $n$ borders and/or punctures. I wish to know the basics of different ...

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### Generalized McShane Identity for Closed Riemann Surfaces

There is an identity for the hyperbolic Riemann surfaces with at least one border. The identity is known as Generalized McShane Identity or Mirzakhani-McShane Identity proved by Mirzakhani in her ...

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### Does the Torelli space appear “in nature”?

What I mean by the (slightly facetious) title is:
The classical theory of algebraic curves from the 19th century was split in two in the 20th century (much like the theory of groups): the theory of ...

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### Embedding Riemann surfaces into $\mathbb P^2$

Suppose I am given a Riemann surface $\Sigma_g$ of genus $g$. What is known about the sufficient and necessary conditions needed on $\Sigma_g$ to have an embedding into $\mathbb P^2$?
If $\mathcal ...

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### Teichmuller Space of a Disk with Holes and Boundary Punctures

If we consider a disk $D$ with $h$ holes and $n$ punctures on the boundary of the disk, then:
Is there a uniformization theorem for such surfaces?
What is the condition on $h$ and $n$ such that we ...

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### Shear coordinates, lambda lengths, cluster variables

I am trying to understand the relations among Shear coordinates, lambda lengths, cluster variables, in the paper. Is the following correct?
Lambda lengths = cluster A-variables
Shear coordinates = ...

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### Weil-Petersson norm of a Beltrami form

I'm reading Scott Wolpert's paper Noncompleteness of the Weil-Petersson metric for Teichmüller Space.
He defines a path leading to the boundary of Teichmüller Space by giving surfaces $R_t$ ...

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### Riemann Theta Function On Hyperbolic Riemann Surfaces

The Riemann theta function for a genus $g$ closed Riemann surface with period matrix $\tau=[\tau_{ij}]$ is defined by
$$\theta(\{z_1,\cdots,z_g\}|\tau)=\Sigma_{n\in\mathbb{Z}^g}e^{\pi i(n\cdot\tau\...

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### If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?

Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...

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### reference - Grothendieck on Thurston's work

In his 'dernières' years Grothendieck gets "interested" in Thurston's work.
"[...] je me suis intéressé ces dernières années - la géométrie hyperbolique à la Thurston et ses relations au groupe de ...

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### Geometric intersection number for product of elements of the fundamental group

Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...

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### Reference request: Families of curves and associated mapping classes

In Teichmüller theory, we consider families of genus $g$ smooth complex projective curves with $n$ distinguished points.
Assume $2g-2+n>0$ and, for convenience, $(g,n)\neq(1,1),(2,0)$.
Denote ${T}...

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### Uniformizations of the bordered/punctured Riemann surfaces

The uniformization theorems of Riemann surfaces state that any Riemann surface can be constructed by an action of some group on some space. It is quite hard to find materials relating different ...

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### Teichmüller space on non-orientable closed surfaces

It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a ...

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### Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space?

It is known that Teichmüller distance ($d_{Teich}$) on Teichmüller space is complete, whereas Weil-Petersson distance ($d_{WP}$) is not complete.
See for example the article
Wolpert, Scott. ...

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### Pseudo-Anosov maps with same dilatation.

Let $S$ be a hyperbolic surface. Suppose $\mathcal{T}$ denotes the Teichmuller space of $S$ and $Mod(S)$ denotes the mapping class group of $S$. Given any pseudo-Anosov element $f\in Mod(S)$, suppose $...

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### Kra's theorem of Pseudo-Anosov maps

Let $S$ be a surface of negative Euler characteristic. Consider the Birman exact sequence:
$$1\xrightarrow{ }\pi_1(S,p)\xrightarrow{P} Mod(S,p)\xrightarrow{ }Mod(S)\xrightarrow{ }1$$
In his paper ...