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4
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0answers
121 views

Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space". The tangent space at ...
6
votes
1answer
152 views

Geometric quantization of Teichmuller space

The quantizations of Teichmuller space I have seen are via special coordinates (e.g. the paper of Chekhov and Fock) or conformal blocks. Does one get an equivalent quantization by geometric ...
2
votes
0answers
104 views

Dessin d'enfant and moduli space of bordered/punctured hyperbolic Riemann surfaces

Belyi's theorem states that if a Riemann surface could be defined as an algebraic curve over an algebraic number field, then this Riemann surface could be described by a Dessin d'enfant. I have two ...
1
vote
0answers
49 views

Change of length of curve when Fenchel-Nielsen length coordinate increase

Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the ...
0
votes
1answer
71 views

Sequence of translation surfaces and length of saddle connections

A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is ...
4
votes
0answers
142 views

Questions on Thurston's metric on Teichmüller space

I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...
4
votes
1answer
121 views

Definition of the Teichmuller space via topological marking of the $\pi_1$

Let $S_0$ be a compact orientable and oriented surface, of genus $g$, fixed once for all. For a fixed point $s_0\in S_0$, one considers a symplectic basis $a_1,\ldots,a_g,b_1,\ldots,b_g$ of $\pi_1(...
1
vote
0answers
24 views

What is the topology of teichmuller space of a given hyperhahler manifold

From verbitsky's paper, I read that the teichmuller space of a given hyperkahler manifold is not hausdorff but it's smooth. I am a bit confused about how to put a natural topology on the teichmuller ...
3
votes
0answers
120 views

Uniform continuity of length function on geodesic currents

I'm starting to study geodesic currents and I have a question concerning uniform continuity. Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...
3
votes
1answer
364 views

Confusion about a result on Shimura and Teichmüller curves

It is shown by M. Moeller (M. Moeller, Shimura- and Teichmüller curves) that there are only 2 Shimura and Teichmüller curves in the moduli space of curves $M_g$, namely, the ones given by $y^4=x(x-1)(...
1
vote
2answers
189 views

Complex structure on a punctured torus giving a complex structure on the torus?

Can anyone provide an idea of the proof or a reference of the fact that a complex structure on the once punctured torus extends to one on the torus? In other words, the Teichmuller space of the ...
4
votes
1answer
192 views

Canonical models of Teichmüller curves

It is well-known that Shimura varieties can be defined over number fields and that moreover they possess canonical model over number fields. On the other hand, Teichmüller curves can also be defined ...
4
votes
1answer
226 views

The hyperbolic metric on a flat surface

Let $S$ be a closed oriented surface of genus $\geq 2$ and $\mathcal{F}_n(S)$ be the space of flat metrics with conic singularities on $S$ whose cone angles are of the form $2k\pi/n$ ($k\in\mathbb{N}$)...
4
votes
1answer
149 views

Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...
2
votes
1answer
42 views

Is the length function associated with the twist parameter an increasing function?

Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is ...
0
votes
0answers
103 views

What is the Beltrami differential?

Let $R,S$ be Riemann surfaces and $f: R \to S$ an orientation preserving diffeomorphism. Then $f$ determines what is called a Beltrami differential denoted by $\mu \frac{d\bar{z}}{dz}$. Local ...
4
votes
0answers
84 views

Barycentric interpolation in hyperbolic triangles

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...
1
vote
0answers
52 views

Complex structure and antipode map on the space of measured geodesic laminations

Fix a closed hyperbolic surface $S$, which represents a point in the Teichmüller space $\mathcal{T}$ of the underlying topological surface. Thurston's earthquake theorem implies an identification ...
4
votes
0answers
76 views

Metrics on Teichmüller spaces

I know that Teichmüller $\mathcal{T}_g$ spaces support different metrics. One of them is the Bergman metric; which is a particular case of the Bergman metric on any domain of holomorphy. On the other ...
5
votes
2answers
110 views

Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of Quasiconformal Surface. Definition: A Quasiconformal surface $S$ is a ...
3
votes
1answer
152 views

Decomposition of hyperbolic surfaces near cusps into annuli

Let $C=\mathbb{H}/\Gamma$ be a hyperbolic surface and $c$ a cusp of this sruface. In the paper "Billiards and Teichmüller curves on Hilbert modular surfaces" by C. McMullen, it is claimed that near ...
5
votes
1answer
275 views

Ivanov's metaconjecture on surface homeomorphisms.

In Fifteen problems about MCG Ivanov stated the following metaconjecture: Every object naturally associated to a surface S and having a sufficiently rich structure has $Mod(S)$ as its groups of ...
9
votes
1answer
526 views

A question about Mirzakhani et. al.'s algebraicity theorem

While the geodesic flow on a complete hyperbolic surface is ergodic, the closure of an individual orbit (a geodesic line) can take a complicated fractal-like shape. Nonetheless, there is an ...
2
votes
1answer
205 views

Reference request: flat surfaces

When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic: The ...
8
votes
2answers
154 views

Limits at infinity of fellow-travelling sequences in Teichmuller space,

I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling. Suppose that we have closed surface of genus $g\geq 2$, and ...
2
votes
1answer
188 views

Torsion elements in the mapping class group

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...
7
votes
2answers
229 views

Teichmuller geodesics vs. geodesics in the hyperbolic plane

Geodesics in $\mathbb H^2$ have the following properties: For every two points in the plane there exists a unique geodesic joining them. Every geodesic determines exactly two points on the ...
1
vote
0answers
81 views

Is triple point intersection 'generic' in Teichmuller space?

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...
4
votes
1answer
121 views

$L^p$ stability of the Beltrami equation

Let's assume that $f$ is a quasiconformal homeomorphism of $\mathbb{C}$ with Beltrami coefficient $\mu = \frac{\bar{\partial} f}{\partial f}$. Notice that by definition $\Vert \mu \Vert _{L^{\infty}} &...
4
votes
1answer
230 views

Angle between geodesics in hyperbolic surface

Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma_1$ and $\gamma_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic ...
3
votes
1answer
190 views

A query about Hatcher flow on arc complex

In the paper "Triangulations of Surfaces" Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes ...
1
vote
1answer
102 views

Reference for the result that the systol map from Teichmuller space to curve complex is coarsely Lipschitz

Let $\mathcal{T}(S)$ denotes the Teichmuller space of a finite type surface $S$ equipped with Teichmuller metric and $\mathcal{C}(S)$ denotes the curve complex. Define a map $$\phi:\mathcal{T}(S)\...
1
vote
2answers
130 views

Does the Teichmüller space of the pair of pants admit a continuous global section?

Let $P$ be a pair of pants, $H(P)$ be the space of smooth hyperbolic Riemannian metrics with geodesic boundary on $P$, and $T(P)$ be the Teichmüller space of $P$ (quotient of $H(P)$ under smooth ...
1
vote
1answer
158 views

Ratner theorem and dense geodesic planes in hyperbolic manifolds

Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
3
votes
1answer
89 views

Distorsion of subgroups of the mapping class group

Let $S_{g,b}$ be an oriented surface with $b$ boundary components and $S_g^b$ be an oriented surface with $b$ punctures. Let $\mathrm{Mod}(S_{g,b})$ and $\mathrm{Mod}(S_g^b)$ their (orientation ...
3
votes
0answers
141 views

$\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$. Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...
1
vote
0answers
51 views

Connectivity and contarctibility of complexes associated to curves and arcs

There are various complexes associated to a surface using the curves and arcs e.g. Curve complex, Arc complex, curve arc complex and so on (for a collection of such objects see This). Now to ...
0
votes
1answer
138 views

A doubt from “Geometry of the complex of curves II: Hierarchical structure” by Masur and Minsky

In the paper "Geometry of the complex of curves II: Hierarchical structure" (Paper) there is a construction of curve complex for an Annular subdomain (2.4). The construction depends on the domain ...
5
votes
1answer
258 views

Link for “A spine for Teichmüller space”, preprint by Thurston

Can someone please give any link or mention any source where I can find the following preprint. W.Thurston, A spine for Teichmüller space, preprint, three pages, 1986.
2
votes
1answer
185 views

Coordinates for Teichmuller space for compact conformal surfaces

Fenchel-Nielsen coordinates give a coordinatization of Teichmuller space for compact conformal surfaces admitting a pants decomposition. But not all compact conformal surfaces (possibly with boundary, ...
0
votes
1answer
92 views

Quasiconformal extensions of diffeomorphisms

Let $\gamma:\mathbb R\to\mathbb R$ be an increasing diffeomorphism. Then it is well known that there exist quasiconformal mappings of the upper half plane which extends $\gamma$. One way to construct ...
4
votes
1answer
153 views

(Un)distorted subgroups in the mapping class group: reference required.

Let $S$ be an orientable surface with negative Euler characteristic. Can somebody provide a reference for the following well-known results: the cyclic subgroup generated by a pseudo-Anosov element ...
0
votes
1answer
217 views

Torelli group of a punctured elliptic curve

Let $T_{g,n}$ be the Torelli group of a $n$-punctured surface $S=\overline{S}\setminus\{x_1,\ldots,x_n\}$, with $\overline S$ orientable, closed and of genus $g$. By definition, $T_{g,n}$ is the ...
1
vote
1answer
151 views

About a definition of quasi-conformal maps

A book I'm reading gives the following definition for quasi-conformal maps: If $f$ is a homeomorphism of a metric space X to itself, $f$ is K-quasi-conformal if and only if for all $z \in X$: ...
2
votes
0answers
63 views

Explicit local expression for Bers embedding in genus 2

Let $\mathcal T_{g,n}$ be the Teichmüller space of genus g compact Riemann surfaces with $n$ marked points. According to Riemann, this is a complex manifold of complex dimension 3g-3+n. Bers ...
1
vote
1answer
36 views

Length of the transversal for surfaces with cusps

In Peter Buser's Geometry and Spectra of Compact Riemann Surfaces he shows that the length of the transverse curve to a geodesic in a pants decomposition on a compact hyperbolic surface has length a ...
1
vote
1answer
177 views

A continuous version of Teichmuller uniqueness

By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and $\infty$...
1
vote
1answer
131 views

Centralizer of a pseudo-Anosov element

What is the centralizer of a pseudo-Anosov element in the mapping class group of an orientable punctured surface? Is it cyclic? If so, where can I find a proof?
1
vote
0answers
79 views

Isomorphism type of mapping class group

Let $MCG(S_{g,b}^s)$ be the mapping class group of a surface $S_{g,b}^s$. Assume that it is not trivial. Is it true that $MCG(S_{g,b}^s)$ is isomorphic to $MCG(S_{g',b'}^{s'})$ if and only if $ S_{g,...
1
vote
1answer
180 views

Quasi-isometric embeddings of the mapping class group into the Teichmuller space

Does there exist a quasi-isometric embedding $$MCG(S) \to (\mathrm{Teich}(S), d)$$ for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?