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3
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81 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
60 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 ...
7
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1answer
117 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....
7
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0answers
96 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 ...
4
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1answer
210 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
56 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 ...
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0answers
22 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 ...
2
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1answer
202 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 ...
0
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0answers
47 views

Fourier coefficients of a variation in Teichmuller theory

Prove that for $\dot w[\mu](\zeta)=-\frac{(\zeta-1)(\zeta+1)(\zeta+i)}{\pi}\left\{\iint_{\Delta} \frac{\mu(z) d x d y}{(z-1)(z+1)(z+i)(\zeta-z)}+\iint_{\Delta} \frac{i \overline{\mu(z)} d x d y}{(\bar{...
13
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3answers
417 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
153 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 ...
1
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0answers
126 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
70 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
66 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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33 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 ...
5
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122 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 ...
3
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105 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
251 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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66 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 ...
4
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3answers
151 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
59 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 ...
4
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2answers
251 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
305 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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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
145 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 ...
20
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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
87 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/\...
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3answers
691 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
118 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
65 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
46 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 ...
2
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1answer
105 views

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 ...
4
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1answer
151 views

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 ...
3
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1answer
105 views

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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0answers
147 views

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 ...
6
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4answers
604 views

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 ...
12
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2answers
362 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 ...
4
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1answer
285 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 $...
6
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1answer
200 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 ...
2
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1answer
89 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 $\tau$ (...
6
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0answers
239 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 ...
9
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0answers
154 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 ...
4
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2answers
175 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 ...
2
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1answer
115 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 $...
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0answers
43 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 ...
4
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1answer
141 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 ...
1
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0answers
88 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 ...
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0answers
127 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 ...
3
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1answer
89 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
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1answer
214 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 $...

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