Questions tagged [teichmuller-theory]

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Fenchel–Nielsen coordinates vs Fock–Goncharov coordinates

Consider an orientable surface $S$ and its Teichmüller space $S$, which is the space of representations of its fundamental group $T(S)=\{\rho: \pi_1(S) \to \operatorname{SL}(2,\mathbb{R})\}$. Fock and ...
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81 views

What is mean of action of group on tower?

What is mean of action of group on tower ?in the page 17 of following paper (action on teichmuller tower ) In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class ...
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1 vote
1 answer
59 views

Measured geodesic laminations have either discrete or Cantor set local cross-sections

I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076. In section 1, after he defines measured geodesic laminations, he makes the ...
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7 votes
0 answers
117 views

The relation between Wolf's and Teichmüller's parametrization of the Teichmüller space

Let $\mathcal{T}_g$ be the Teichmüller space of Riemannian surface structures on an oriented 2-dimensional manifold of genus $g$. Fix a point $S \in \mathcal{T}_g$. There are two different ways to ...
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2 votes
0 answers
60 views

Automorphism group of moduli space $M_{0,n}$

Statement: the group of complex automorphisms of the moduli space $M_{0,n}$ of complex $n$-marked genus 0 curves is isomorphic to $\mathfrak S_n$: one has ${\rm Aut}(M_{0,n})=\mathfrak S_n$ I believe ...
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3 votes
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57 views

Behavior of the period map at the boundary of the Teichmuller space

Fix a compact oriented surface $S$ of genus $g$. Any complex structure $J$ on $S$ gives by the Hodge decomposition a linear complex structure $J'$ on $H_1(S,\mathbb{R})$. The map $J\mapsto J'$ is a ...
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2 votes
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58 views

Equicontinuity, Beltrami coefficients and Sequence of top/bottom semi-annuli

In the Beltrami equation literature, one approach to showing equicontinuity for pairs $(f_{n},\mu_{n})$ (where $\partial_{\bar{z}}f_{n}=\mu_{n}(z)\partial_{z}f_{n}$) is via the relations of moduli and ...
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177 views

What is the topology of the Hitchin component?

Let $S$ be a closed orientable connected surface of genus $g\geq2$. In Lie groups and Teichmüller space Hitchin shows that one of the component of $\chi(S)=\operatorname{Hom}^{+}(\pi_1(S),\...
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  • 443
5 votes
2 answers
232 views

Fenchel-Nielsen length-length coordinates on Teichmueller space?

Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...
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15 votes
2 answers
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What do the components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ look like?

Let $S$ be a closed orientable surface of genus at least $2$. I'm interested in the connected components of $\operatorname{Hom}(\pi_1(S),\operatorname{SL}_n(\mathbb{R}))$ for $n$ at least $3$. I know ...
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7 votes
3 answers
210 views

Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...
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310 views

Integration à la Mirzakhani

Let $$ \gamma = \sum_i c_i \gamma_i $$ be a multi-curve on a hyperbolic surface $S$. For any $f: \mathbb{R}^+ \to \mathbb{R}^+$ one can define $$ f_\gamma (X) = \sum_{\alpha \in \mathrm{Mod} . \gamma} ...
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2 votes
1 answer
153 views

Two definitions of Teichmüller space: relative isotopy or not?

The definition of Teichmüller space on wikipedia via marked Riemann surfaces say that two markings are equivalent if the map $fg^{-1}$ is isotopic to a holomorphic diffeomorphism. The definition on ...
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"Pushforward" of a universal curve by a map (Riemann surfaces)?

For a Riemann surface $X$ with a Beltrami form $\mu \in M(X),$ ($M(X)$ is the space of Beltrami forms on the Riemann surface), we can define the Riemann surface $X_\mu.$ In John H. Hubbard's ...
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1 vote
1 answer
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Motivation for the definition of $L^p$ norm for quadratic and Beltrami differentials

According to Riemann surfaces, dynamics and geometry by C. McMullen (Course notes), the definition for a quadratic differential $\phi$ on a Riemann surface $X$ is given by $$ \|\phi\|_p = \left(\...
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2 votes
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Beurling’s extremality criterion for curves: two versions

I see Beurling’s extremality criterion at two places: the proof is almost identical, but the statement is very different. Below, $$ \ell_\rho (\gamma) = \int_\gamma \rho(z) |dz|. $$ "Extremal&...
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3 votes
0 answers
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Non commutative Teichmuller theory

Perhaps the first example in Teichmuller theory is the following proposition: Proposition: Let $1<r<R$. Then two annular region $U_r=\{z\in \mathbb{C}\bigm|1<|z|<r\}$ and $U_R=\{z\in \...
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3 votes
2 answers
205 views

Learning roadmap for Lorentzian geometry

I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF. I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (...
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7 votes
1 answer
355 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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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 ...
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2 votes
2 answers
224 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&...
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3 votes
0 answers
88 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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80 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 ...
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2 votes
0 answers
55 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 $\...
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2 votes
2 answers
181 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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1 vote
0 answers
35 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 $(G,L,e)$ ...
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3 votes
1 answer
214 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 ...
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4 votes
0 answers
67 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) = \...
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9 votes
1 answer
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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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1 vote
0 answers
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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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1 vote
0 answers
111 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 ...
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2 votes
0 answers
40 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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1 vote
0 answers
70 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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  • 1,643
6 votes
2 answers
220 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 ...
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6 votes
1 answer
133 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 ...
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4 votes
0 answers
95 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 ...
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4 votes
1 answer
576 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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3 votes
0 answers
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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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4 votes
0 answers
91 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 ...
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11 votes
1 answer
322 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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7 votes
0 answers
150 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 ...
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  • 191
6 votes
1 answer
310 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 ...
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2 votes
1 answer
94 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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4 votes
1 answer
164 views

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 ...
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3 votes
1 answer
385 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 ...
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14 votes
3 answers
528 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 $...
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2 votes
1 answer
188 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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  • 273
1 vote
0 answers
129 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 ...
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  • 167
3 votes
1 answer
80 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 ...
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  • 273
3 votes
0 answers
74 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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