Questions tagged [teichmuller-theory]
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251
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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 ...
3
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100
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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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0
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71
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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 $\...
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3
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339
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Why a Teichmüller map is not a pseudo-Anosov?
Let $X$ be a Riemannian surface. Suppose $f:X\to X$ is a Teichmüller 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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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 $(G,L,e)$ ...
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275
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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 ...
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85
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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) = \...
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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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49
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"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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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 ...
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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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0
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87
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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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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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1
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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 ...
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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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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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101
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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 ...
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105
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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 ...
14
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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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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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1
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380
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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 ...
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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 ...
5
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228
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Explicit check of the invariance of the Weil-Petersson form
Using Fenchel-Nielsen coordinates, the Weil-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 of ...
3
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1
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709
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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 ...
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3
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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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1
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227
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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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132
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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 ...
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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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0
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82
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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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39
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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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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 ...
4
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281
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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 ...
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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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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 ...
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3
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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 ...
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276
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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 ...
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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}\...
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1
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384
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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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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 $f(\infty)=Q$}\},...
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1
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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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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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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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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 ...
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168
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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}^{\...
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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 $l,h>0$. ...
3
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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 ...
2
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1
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121
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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 ...
3
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1
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148
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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 ...