Questions tagged [teichmuller-theory]

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13 votes
2 answers
471 views

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 ...
Dylan Thurston's user avatar
4 votes
1 answer
505 views

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 $...
Alex's user avatar
  • 41
6 votes
1 answer
297 views

Compactifications of SL(2)-character varieties of surfaces

Thurston compactified the Teichmüller 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 ...
Adam's user avatar
  • 2,370
2 votes
1 answer
101 views

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 $\tau$ (...
Adam's user avatar
  • 2,370
7 votes
0 answers
378 views

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 ...
user avatar
9 votes
0 answers
166 views

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 ...
F. Germano's user avatar
4 votes
2 answers
309 views

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 ...
Cusp's user avatar
  • 1,703
2 votes
1 answer
126 views

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 $...
cooper90's user avatar
1 vote
0 answers
46 views

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 ...
giulio bullsaver's user avatar
5 votes
1 answer
238 views

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 ...
Adam's user avatar
  • 2,370
1 vote
0 answers
108 views

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 ...
Adam's user avatar
  • 2,370
2 votes
1 answer
192 views

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 ...
Adam's user avatar
  • 2,370
3 votes
1 answer
119 views

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 ...
Adam's user avatar
  • 2,370
2 votes
1 answer
251 views

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 $...
Angy's user avatar
  • 61
3 votes
0 answers
188 views

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 ...
user avatar
2 votes
1 answer
232 views

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 $...
Cusp's user avatar
  • 1,703
9 votes
2 answers
358 views

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 ...
Adam's user avatar
  • 2,370
5 votes
1 answer
213 views

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 $\...
Dmitri Gekhtman's user avatar
5 votes
1 answer
191 views

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 \...
QGravity's user avatar
  • 969
4 votes
1 answer
276 views

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 ...
asv's user avatar
  • 21.1k
7 votes
1 answer
818 views

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" ...
asv's user avatar
  • 21.1k
12 votes
2 answers
1k views

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 ...
asv's user avatar
  • 21.1k
3 votes
1 answer
735 views

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 ...
QGravity's user avatar
  • 969
3 votes
0 answers
114 views

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 ...
Thomas Kojar's user avatar
  • 4,414
3 votes
0 answers
99 views

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, ...
xdyj's user avatar
  • 31
5 votes
0 answers
148 views

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,...
Cusp's user avatar
  • 1,703
2 votes
0 answers
228 views

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 ...
QGravity's user avatar
  • 969
1 vote
0 answers
321 views

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 ...
QGravity's user avatar
  • 969
5 votes
0 answers
297 views

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 ...
Nati's user avatar
  • 1,971
6 votes
1 answer
608 views

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 ...
Harry Reed's user avatar
2 votes
0 answers
434 views

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 ...
QGravity's user avatar
  • 969
2 votes
0 answers
139 views

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 = ...
Jianrong Li's user avatar
  • 6,101
3 votes
0 answers
166 views

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 $R_t$ ...
Patrick Haggerty's user avatar
6 votes
2 answers
466 views

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\...
QGravity's user avatar
  • 969
0 votes
2 answers
216 views

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 ...
Anubhav Mukherjee's user avatar
15 votes
0 answers
1k views

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 ...
tttbase's user avatar
  • 1,700
3 votes
0 answers
405 views

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 ...
Cusp's user avatar
  • 1,703
2 votes
0 answers
95 views

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}...
gcousin's user avatar
  • 306
6 votes
2 answers
1k views

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 ...
QGravity's user avatar
  • 969
8 votes
2 answers
575 views

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 ...
Curious's user avatar
  • 81
8 votes
1 answer
368 views

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. ...
user197284's user avatar
8 votes
2 answers
534 views

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 $...
Cusp's user avatar
  • 1,703
4 votes
0 answers
233 views

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 ...
Cusp's user avatar
  • 1,703
1 vote
1 answer
302 views

Homotopy in Teichmüller space definition: to be or not to be? That is the question

In the book Introduction to Teichmüller Spaces, by Imayoshi & Taniguchi, we finde the following definition of the Teichmüller space of a Riemann surface $R$, denoted $T(R)$: I want to draw ...
Derso's user avatar
  • 113
1 vote
0 answers
62 views

Quasiconformal constant in Nielsen isomorphism theorem

Let $\rho_1$ and $\rho_2$ be two faithfull and discrete representations of the fundamental group of a compact surface into $PSL(2,\mathbb{R})$. The Nielsen isomorphism theorem says that there exists a ...
François Fillastre's user avatar
7 votes
1 answer
365 views

Selberg Zeta Function and Fenchel-Nielsen Coordinates

According to Uniformization theorem every compact Riemann surface $\Sigma$ of genus $g\ge2$ is isomorphic to a space that can be obtained by the action of a Fuchsian group on upper half plane $\mathbb{...
QGravity's user avatar
  • 969
4 votes
0 answers
140 views

Does uniform convergence of (Riemannian) distances implies convergence of conformal structures?

I don't know much about the Teichmüller space, so maybe the question I ask is well known; still I can not find the answer by myself... Let $\Sigma$ be a closed surface. Let $g_m$ be a sequence of (...
Clem.'s user avatar
  • 353
2 votes
0 answers
162 views

Convexity of length function for surfaces with boundary

In the paper "The Nielsen realization problem" (here), Kerckhoff proved that the length function on the Teichmüller for closed surface is convex. In his paper "Geodesic length functions and the ...
Cusp's user avatar
  • 1,703
5 votes
1 answer
284 views

Proof that the length function $\ell: \operatorname{Teich}(S) \to \mathbb{R}^\mathcal{S}$ is injective without the $9g-9$ theorem

In Chapter 10 about Teichmüller spaces of Farb and Margalit's "A Primer to Mapping Class Groups", the length function $$\ell: \operatorname{Teich}(S) \to \mathbb{R}^\mathcal{S}$$ is described, where $...
Huy's user avatar
  • 243
3 votes
1 answer
147 views

Non-lattice Veech groups

I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}_2(\mathbb{R})$. They seem to have been ...
Selim G's user avatar
  • 2,626