Questions tagged [teichmuller-theory]
The teichmuller-theory tag has no usage guidance.
250
questions
13
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2
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471
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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 ...
4
votes
1
answer
505
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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 $...
6
votes
1
answer
297
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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 ...
2
votes
1
answer
101
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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 $\tau$ (...
7
votes
0
answers
378
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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 ...
9
votes
0
answers
166
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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 ...
4
votes
2
answers
309
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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 ...
2
votes
1
answer
126
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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 $...
1
vote
0
answers
46
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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 ...
5
votes
1
answer
238
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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 ...
1
vote
0
answers
108
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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 ...
2
votes
1
answer
192
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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 ...
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 ...
2
votes
1
answer
251
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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 $...
3
votes
0
answers
188
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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 ...
2
votes
1
answer
232
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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 $...
9
votes
2
answers
358
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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 ...
5
votes
1
answer
213
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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
$\...
5
votes
1
answer
191
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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 \...
4
votes
1
answer
276
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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 ...
7
votes
1
answer
818
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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" ...
12
votes
2
answers
1k
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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 ...
3
votes
1
answer
735
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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 ...
3
votes
0
answers
114
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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 ...
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, ...
5
votes
0
answers
148
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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,...
2
votes
0
answers
228
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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 ...
1
vote
0
answers
321
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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 ...
5
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0
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297
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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 ...
6
votes
1
answer
608
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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 ...
2
votes
0
answers
434
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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 ...
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 = ...
3
votes
0
answers
166
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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$ ...
6
votes
2
answers
466
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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\...
0
votes
2
answers
216
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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 ...
15
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0
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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 ...
3
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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 ...
2
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0
answers
95
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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}...
6
votes
2
answers
1k
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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 ...
8
votes
2
answers
575
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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 ...
8
votes
1
answer
368
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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. ...
8
votes
2
answers
534
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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 $...
4
votes
0
answers
233
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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 ...
1
vote
1
answer
302
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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 ...
1
vote
0
answers
62
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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 ...
7
votes
1
answer
365
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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{...
4
votes
0
answers
140
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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 (...
2
votes
0
answers
162
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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 ...
5
votes
1
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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 $...
3
votes
1
answer
147
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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 ...