Questions tagged [teichmuller-theory]

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6
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1answer
310 views

Paths $tg_1+(1-t)g_0$ in the moduli space of Riemann surfaces

Suppose $S$ is a smooth compact oriented surface without boundary. Let $g_0$ and $g_1$ be two smooth Riemannian metrics on $S$. Consider the interpolating path of metrics $g_t=g_1t+g_0(1-t)$. Recall ...
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153 views

References on Hyperbolic Geometry and Teichmuller Theory

I am asking a soft question here. I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
2
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2answers
182 views

References on Riemann surfaces

I have asked the question in MSE, but did not get an answer. I am asking a soft question here. I am interested in learning about Hyperbolic Geometry. I have read the book named "Fuchsian Groups&...
3
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0answers
74 views

Clarifications about a proof of (the measurable Riemann) mapping theorem in Hubbard's book on Teichmuller theory,

On page 151 of Hubbard's book, the author is proving the following theorem( Prop.4.6.2 ): Suppose $\mu$ is a real analytic function on a domain $U$ of $\mathbb{C}$. Then every $z \in U$ has a ...
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0answers
65 views

Relating different parametrizations of moduli space of Riemann surfaces

I would like to understand, as explicitly as possible, how different coordinates on the moduli space of Riemann surfaces are related: On the one hand, there is a parametrization coming from hyperbolic ...
2
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0answers
44 views

Thurston measure of Dehn-Thurston ball center at a multi curve

Given a surface of genus $g$ with $n$ singularities, and a decompostion of pair of pant $P=(p_1,...,p_{3g-3+n})$ one can give coordinate (called Dehn-Thurston coordinate) on the space of lamination $\...
2
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2answers
153 views

Why a Teichmuller map is not a pseudo-anosov?

Let $X$ be a riemannian surface. Suppose $f:X\to X$ is a Teihmuller map with respect to a quadratic differential $q$ on $X$. This means that, if $q=dz^2$ in local coordinates in a neighborhood of ...
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0answers
29 views

flips on labelled fatgraphs and mapping classes

A fatgraph $G$ is a graph with a cyclic ordering of the edges at each vertex. A labelled fatgraph $(G,L)$ is a fatgraph together with a labelling $L$ of each edge. A labelled fatgraph spine $(G,L,e)$ ...
3
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1answer
162 views

Representation of the mapping class group in terms of flips on triangulations

$\DeclareMathOperator{\MCG}{\operatorname{MCG}}$Consider a bordered, punctured, orientable surface $S$. Associated to it there is its mapping class group $\MCG(S)$. One way to concretely think about ...
3
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0answers
53 views

Harmonic maps versus Teichmuller maps between Riemann surfaces

Let $(X,\phi)$ be an element of Teichmuller space $\cal T_g$ and $q$ a (holomorphic) quadratic differential on $X$. Teichmuller geodesic flow gives a family of marked Riemann surfaces $(Y_t,\psi_t) = \...
9
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1answer
122 views

When do the lengths of simple closed curves determine a hyperbolic surface?

Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$...
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32 views

“Convergence” of bordered Riemann surfaces to a congruence surface

Let $\Gamma(N)$ be the principal congruence subgroup of level $N\geq 3$, $H$ the upper half-plane and $C(N)=H/\Gamma(N)$ be the corresponding Riemann surface. In his paper " Congruence Subgroups ...
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81 views

Well-definedness of marking a Riemann surface by diffeomorphisms in the context of Teichmüller spaces

In "An introduction to Teichmüler Theory" of Yoichi Imayoshi and Masahiko Taniguchi the Teichmüller space is defined as follows: fix a compact Riemann surface $R$ of genus $g$, a marking on ...
2
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0answers
37 views

Whether or not two distinct points in Teichmuller space induce absolutely continuous volume forms on the unit tangent bundle of a surface?

Let $S$ be a closed orientable surface of genus greater than two. Let $g$ and $g'$ be metrics two of constant curvature. I guess we an think of these as two points in the Teichmüller space $\mathcal{T}...
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0answers
51 views

Weil-Petersson metric with respect to covering

Let $S$ be a closed oriented surface of genus $g\geq 2$. Consider the Teichmuller space $T(S)$. Let $d_t$ be the Teichmuller metric and $d_{WP}$ be the Weil-Petersson metric on $T(S)$. Let $P:S_1\...
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0answers
100 views

Anosov structure of Fuchsian representations

I was reading a paper by Labourie https://arxiv.org/pdf/math/0401230.pdf and I'm having trouble understanding a proof. First, here's the setup. $G = PSL(n,\mathbb{R})$ and $U$ is the subgroup of ...
6
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2answers
170 views

Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$Let $S$ be a compact oriented surface with nonempty boundary. There are two variants of Teichmuller space for $S$ you might consider: The ...
6
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1answer
104 views

Number of Fuchsian groups with same trace field

Let $\Gamma,\Sigma\subset \mathrm{SL}_2({\mathbb R})$ be cocompact arithmetic subgroups. They are called commensurable in the wider sense, if there exists $g\in \mathrm{SL}_2({\mathbb R})$, such that ...
4
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0answers
82 views

Holomorphic maps on moduli space and Deformation theory

Let $\mathcal{M},\mathcal{F}$ be the classifiying spaces (i.e. complex manifolds) of two (possibly) different moduli problem. To give a map $$f:\mathcal{M}\rightarrow \mathcal{F}$$ means that for each ...
4
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1answer
446 views

The (measurable) Riemann mapping theorem

The Riemann mapping theorem says that a strict, nonempty open subset of the complex plane is conformally equivalent to the unit disk. The measurable Riemann mapping theorem asserts the existence and ...
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0answers
92 views

Isomorphism between two families of curves over the Teichmueller space

In his construction of the Teichmueller space of curves of genus $\geq 2$ Grothendieck states in Corollaire 2.4 that the map $$\underline{Isom}_S(X,Y) \xrightarrow{} S$$ is finite. The map represents ...
4
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0answers
73 views

Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...
11
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1answer
293 views

Cutting up the Bring surface into six pairs of pants

The Bring sextic, with 120 automorphisms, is the numerically most symmetric compact Riemann surface of genus 4. To cut it up into six pairs of pants, we need to cut along nine disjoint geodesic loops....
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120 views

Purely analytic proof of the Nielsen-Thurston classification theorem

I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory and Applications to ...
6
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1answer
265 views

Mirzakhani's hyperbolic method generalized to moduli space of stable maps

I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have ...
2
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1answer
82 views

Coordinates for Laminations: geometric versus shear

Let $S$ be an orientable surface with a triangulation T. A lamination $\ell$ is a simple closed curve on $S$, up to isotopy. We will assume that $\ell$ is drawn in such a way that it intersects the ...
4
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1answer
86 views

Explicit check of the invariance of the Weyl-Petersson form

Using Fenchel-Nielsen coordinates, the Weyl-Petersson metric can be written as $\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$ where $i$ is an index labelling the curves of a pants decomposition ...
3
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1answer
263 views

Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures

In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...
14
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3answers
485 views

Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?

Let $F$ be a compact oriented surface and $\rho:\pi_1(F)\rightarrow SL_2\mathbb{C}$ be a representation. Does there exist a compact oriented three-manifold $M$ with $\partial M=F$ and a homomorphism $...
2
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1answer
164 views

What is the Teichmuller metric on the Teichmuller space of a closed surface of genus 1?

Howard Masur's research asserts that if $S_g$ is a closed surface of genus $g\geq2$, then the Teichmuller space $T(S_g)$ does not have nonpositive curvature. His proof relies on the existence of ...
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0answers
128 views

Area of balls on flat surfaces

Let $S$ be a closed surface of genus $g \geq 2$. Define $\mathrm{Flat}(S)$ to be the set of marked flat metrics on $S$ with cone angles $2\pi+k\pi$ for $k\geq 0$. It is well-known that these all come ...
3
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1answer
71 views

Number of curves in an admissible system of Jordan curves on a surface

Consider a compact Riemann surface of genus $g\geq2$. An admissible system of Jordan curves is a finite collection of Jordan curves $\{\gamma_1,\cdots,\gamma_n\}$ such that they are nonintersecting ...
3
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0answers
68 views

How does one prove that the Teichmuller space of a closed Riemann surface of genus $\geq2$ is uniquely geodesic?

I am reading Masur's paper On a class of geodesic in Teichmuller space. He mentions that $T(S_0)$ where $S_0$ is a closed Riemann surface $g\geq2$ is straight, i.e. uniquely geodesic. It seems a well-...
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0answers
34 views

Effect of plumbing a surface on the marked length spectrum

First I'll recall the plumbing procedure. Let $M$ be a noded Riemann surface with nodes $p_1,\dots, p_n$. There is a family of pairwise disjoint neighbourhoods of each node $U_i$ that has coordinates ...
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0answers
142 views

Is there an equivariant simplicial deformation retract of Teichmüller space?

Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...
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0answers
132 views

Bers' simultaneous uniformization

I have been trying to understand Bers' famous paper "Simultaneous Uniformization". Regarding this paper I have a few questions. Any kind of help will be appreciated. Let $S$ and $S^{'}$ be two ...
4
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1answer
259 views

Degenerate Beltrami equation

Question: Let $\mu:\mathbb C\to \mathbb C$ be a $C^\infty$ function satisfying $|\mu|\le 1$. Let us furthermore assume that the function $\mu$ never takes the value $-1$. Does there exist a $C^\infty$ ...
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0answers
70 views

Bound on the distance from points to the boundary of a hyperbolic surface

Fix $\epsilon\in\mathbb{R}_{>0}$, $\Sigma$ a surface with boundary and let $\mathcal{T}_{\Sigma}(L_{1},...,L_{n})$ denote the Teichmüller space of hyperbolic structures of $\Sigma$ with geodesic ...
5
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3answers
174 views

Teichmuller space for surface with cone points

Working on my current research problem, Teichmuller spaces for surfaces with cone points have come into play. It's fairly easy to formulate some of the definitions, a few basic results, etc. To be ...
4
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0answers
61 views

Converse to Wolpert's Lemma

Recall Wolpert's lemma: Let X,Y be hyperbolic surfaces and $f:X\to Y$ a $K$-quasiconformal homeomorphism. For any homotopy class of curves $c$ let $\ell(c)$ denote the length of the geodesic in the ...
5
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2answers
266 views

Are symplectomorphisms of Weil–Petersson symplectic form induced from surface diffeomorphisms?

Let $S$ be a closed hyperbolic surface of genus $g\geq 2$. Let $(\mathcal{T},\omega)$ be the corresponding Teichmuller space with the Weil–Petersson symplectic from $\omega$. Let $\Phi:\mathcal{T}\...
7
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1answer
321 views

To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need?

To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in,...
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0answers
125 views

Space of biholomorphic maps into a Riemann surface

Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space $$X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},...
1
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1answer
151 views

Lie bracket on the complex valued functions of the space of representations of a Riemann surface

Let $S$ be a closed surface and $G$ be a reductive Lie group. Goldman (Invariant functions on Lie groups and Hamiltonian flows of surface group representations) proved that, for a fairly general class ...
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1answer
1k views

Does anybody do $p$-adic Teichmüller theory?

In "Foundations of $p$-adic Teichmüller theory", Mochizuki describes a theory one of whose goals (according to the author) is to generalize Fuchsian uniformization of Riemann surfaces to the $p$-adic ...
2
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0answers
94 views

Deck transformations of Teichmuller space as a universal cover of Torelli space

I'm reading the article by Geoffrey Mess The Torelli groups for genus 2 and 3 surfaces (pp. 785 - 786), and I'm trying to understand the part that concerns genus 3. We've got a map $UT(S) \to T_3/\...
9
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3answers
742 views

What is the official definition of $\mathcal{M}_g$ as an orbifold, and how much can I ignore it?

There is a well-known description of $\mathcal{M}_g$ as $\mathcal{T}_g/\Gamma$ where $\mathcal{T}_g$ is the Teichmuller space and $\Gamma$ is the mapping class group. Teichmuller space is homeomorphic ...
2
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1answer
143 views

Injectivity of the simple closed curves under geometric intersection number

Let $\Sigma$ be a closed surface of genus $g\geq 2$ and $\mathcal{C}$ be the set of all free homotopy classes of simple closed curves in $\Sigma$. Define $i:\mathcal{C}\rightarrow \mathbb{R}^{\...
3
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0answers
73 views

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 $l,h>0$. ...
3
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0answers
48 views

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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