Skip to main content
MathOverflow

Questions tagged [teichmuller-theory]

Ask Question

The tag has no summary.

265 questions
Newest Active Bountied Unanswered
Filter by
Sorted by
Tagged with
1 vote
0 answers
62 views

Let $S_g$ be a compact orientable surface of genus $g \geq 2$. The Teichmüller space $\mathcal{T}(S_g)$ is defined as the set of equivalence classes of pairs $(X, f)$, where: $X$ is a Riemann surface,...
Framate's user avatar
  • 111
3 votes
1 answer
305 views

Let $T_{g,b}$ be the Teichmüller space of $\mathbb{C}$-curves $\Sigma_{g,b}$ with genus $g$ with $b$ marked points. The end of Section 4 of Drinfeld's famous "On Quasitriangular Quasi-Hopf ...
Qwert Otto's user avatar
  • 967
6 votes
1 answer
189 views

Let $\Gamma$ be a Fuchsian group such that $\mathbb H^2/\Gamma$ is topologically a closed surface $S$. Bonahon notably introduced the space of geodesic currents $C(\Gamma)$ as the space of $\Gamma$-...
Roman's user avatar
  • 443
3 votes
1 answer
180 views

I am learning Chapter 8 of the book A Primer on Mapping Class Groups by Farb and Margalit, and I have a question regarding the paragraph "once-punctured versus closed" on page 235. Let me ...
user560628's user avatar
  • 345
3 votes
0 answers
157 views

Let $\bar{X}$ be a compact Riemann surface of genus $g$, and let $D:=\{p_{1},p_{2},\cdots,p_{n}\}$ be $n$ distinct points on $\bar{X}$. Define $X:=\bar{X}-D$ to be the punctured Riemann surface given ...
Yu Feng's user avatar
  • 401
5 votes
1 answer
374 views

Theorem (Sullivan). Every Fatou component $U$ of $f$ rational map is eventually periodic, that is, there exist $n > m > 0$ such that $f^n(U) = f^m(U)$ I am familiar with the proof: spread around ...
Ricky Simanjuntak's user avatar
  • 123
1 vote
1 answer
263 views

Let $\Sigma$ be a surface of finite type. Let $\mathcal{S}$ be the set of non-trivial isotopy classes of simple closed curves on $\Sigma$. One denotes by $l_x(\alpha)$ the infimal length of curves in ...
Adam's user avatar
  • 1,045
6 votes
0 answers
141 views

In Hitchin's famous paper[1] on the self-dual Yang-Mills equations, he discussed the relation between the stable Higgs bundles and the Teichmüller space for a compact Riemann surface. Namely, through ...
Yongmin Park's user avatar
  • 373
5 votes
1 answer
178 views

Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...
W.Smith's user avatar
  • 275
5 votes
1 answer
247 views

Let $\mathcal{T}_A$ be a Teichmüller space of the sphere $S^2$ with a finite set $A$ of marked points, and suppose that $f \colon \mathcal{T}_A \to \mathcal{T}_A$ is a holomorphic map that has a ...
A B's user avatar
  • 51
4 votes
1 answer
216 views

Let $R$ be a compact Riemann surface of genus $\geq 1$, and let $\omega_1,\ldots,\omega_g$ be holomorphic one forms that form the dual basis of canonical homology basis. Let $(\pi)_{ij}$ be the ...
François Fillastre's user avatar
  • 185
7 votes
2 answers
459 views

Let $\Sigma$ be a finite-type orientable surface with negative Euler characteristic, and $\mathrm{Mod}(\Sigma)$ denote the mapping class group. What are the finite normal subgroups in $\mathrm{Mod}(\...
user avatar
11 votes
0 answers
241 views

There are well-known symplectic forms on the Teichmuller space $\mathcal{T}(\Sigma)$ of a closed surface $\Sigma$ (Wolpert gave a formula in Fenchel-Nielsen coordinates) and the space of measured ...
Ian Agol's user avatar
  • 71.3k
1 vote
1 answer
241 views

I have a rather straightforward and perhaps somewhat naive question: Is there a Teichmüller theory for open surfaces? My motivation basically is that I would like to find out more about the "...
M.G.'s user avatar
  • 7,933
4 votes
1 answer
408 views

What is a good reference for Teichmuller spaces of punctured surfaces $S_{g,n}$ where $n>0$? I am looking for a reference where there is the correct statement and or proof of say the Bers embedding,...
Chitrabhanu's user avatar
  • 737
3 votes
1 answer
207 views

By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space. It is known that the boundary of a Teichmüller domain is ...
Mahdi Teymuri Garakani's user avatar
  • 605
2 votes
1 answer
207 views

Let $S_{g,n}$ denote a genus $g$ surface with $n$ punctures. There is a map $F$ from the Teichmüller space of the punctured surface $T(S_{g,n})$ to the Teichmüller space of the compact surface $T(S_{g}...
Yousuf Soliman's user avatar
  • 343
2 votes
1 answer
247 views

I'm reading Andersen and Kashaev's A TQFT from quantum Teichmüller theory and the following condition in their definition of admissible oriented triangulated pseudo $3$-manifold confused me: ...
Shana's user avatar
  • 247
8 votes
1 answer
901 views

I'm confused about the exact multiplicative factor that relates Goldman symplectic form on the $\operatorname{SL}(2,\mathbb R)$-character variety and the Weil–Petersson symplectic form on Teichmüller ...
AMath91's user avatar
  • 48
0 votes
1 answer
312 views

Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
Mahdi Teymuri Garakani's user avatar
  • 605
2 votes
1 answer
287 views

Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
Mahdi Teymuri Garakani's user avatar
  • 605
7 votes
1 answer
1k views

Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$. Question: is $\mathcal{L}(X)$ a ...
Andrey Ryabichev's user avatar
  • 1,095
1 vote
0 answers
97 views

Consider a sphere with $n$ punctures. If you pick a holomorphic cotangent vector at each puncture, you can canonically construct a holomorphic top form in the corresponding moduli space. (The specific ...
Charles Wang's user avatar
  • 563
1 vote
1 answer
105 views

Let $R$ be a finite Riemann surface (having negative Euler Characteristic) without boundary (may have punctures) and $q$ be a unit area quadratic differential on $R$. We define $\mathcal{MF}_{1}=\{F \...
W.Smith's user avatar
  • 275
3 votes
1 answer
305 views

This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand: $$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
Kenny S's user avatar
  • 87
3 votes
1 answer
606 views

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\char{char}$I am reading a note on Teichmüller space, and I come across a somewhat algebraic problem in the picture below,...
Kenny S's user avatar
  • 87
5 votes
1 answer
599 views

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$ I have a reference request for a proof for the following statement in the title: The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
Chaitanya Tappu's user avatar
  • 153
0 votes
1 answer
218 views

In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as $$g = \frac{...
JMK's user avatar
  • 453
2 votes
1 answer
135 views

Let $S$ be a closed real surface having two complex structures $c_1$ and $c_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which ...
Dorian's user avatar
  • 625
3 votes
0 answers
156 views

In An Introduction to Teichmüller spaces by Imayoshi and Taniguchi, they present in section 6.1.3 the Bers embedding as a map from Teichmüller space of a Riemann surface $X$ to the space of quadratic ...
Jacques's user avatar
  • 605
1 vote
0 answers
101 views

Let $(R,p_1,…,p_n)$ be a Compact Riemann surface of genus $g$ with $n$ marked points. Its deformation space is $H^1(R, K_R^{-1}\otimes\mathcal{O}(-p_1),…,\mathcal{O}(-p_n))$. From Riemann-Roch theorem,...
CharlieHo's user avatar
  • 61
1 vote
0 answers
126 views

Let $X$ be a topological surface of finite type and $\mathcal{T}_X$ be the corresponding Teichmüller space. Let $B$ be a ball with respect to the Teichmüller metric on $\mathcal{T}_X$ (i.e., the ...
A B's user avatar
  • 51
2 votes
1 answer
231 views

For a closed Riemann surface $\Sigma$ of genus $g \geq 2$, the space of holomorphic quadratic differentials on $\Sigma$ can be identified with the cotangent space $T_\Sigma^* \mathcal{T}_g$: in other ...
Leo Moos's user avatar
  • 5,183
4 votes
0 answers
203 views

Let $\Sigma$ be a closed orientable surface of genus $g$. It is well known that every almost complex structure on a surface is induced by a complex atlas. Therefore, if we call $\mathcal{J}(\Sigma)$ ...
Joaquin Lema's user avatar
  • 415
2 votes
1 answer
264 views

There has been a question on the same subject, but I'm asking about something more specific. In the Fenchel–Nielsen coordinates, the Teichmüller space of genus $g$ is represented as $\mathbb{R}^{3g-3}\...
Yuxiao Xie's user avatar
  • 365
2 votes
0 answers
232 views

A discrete, faithful representation of a surface group $G:\pi_1(S_g) \to PSL_2(\mathbb{R})$ is determined, up to conjugacy by $PGL_2(\mathbb R)$, among such representations by the squares of traces of ...
RegularGraph's user avatar
  • 143
1 vote
0 answers
115 views

I'm learning about meromorphic (!) quadratic differentials on Riemann surfaces, and would like to determine the closed trajectories [EDIT: I mean closed geodesics, not just closed trajectories; ...
TSBH's user avatar
  • 121
5 votes
0 answers
191 views

Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...
Atlas Tasilli's user avatar
  • 61
2 votes
0 answers
128 views

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$. Now I am planning to ...
user avatar
2 votes
1 answer
475 views

Let $\Sigma$ be a Riemann surface (not necessarily compact), and $x_1, \cdots, x_k$ a set of points on $\Sigma$. Let $n_1, \cdots, n_k$ be a sequence of integers, each of which is $\geq 2$, and such ...
Josh Lam's user avatar
  • 254
68 votes
6 answers
58k views

Today, somebody posted on the nLab a link to Kirti Joshi's preprint on the arXiv from last month: https://arxiv.org/abs/2210.11635 In that preprint, Kirti Joshi claims that he agrees with Scholze and ...
user avatar
4 votes
0 answers
221 views

Per here the first hints of moonshine appeared around 1974 when Andrew Ogg noticed that quotienting the hyperbolic plane by normalizers of the Hecke Congruence subgroups $\Gamma_{0}(p)$ has genus zero ...
Sidharth Ghoshal's user avatar
  • 3,169
2 votes
1 answer
200 views

Let $S_{g}$ be a genus $g$ closed Riemann surface. The Teichmüller space $\mathcal{T}(S_{g})$ is the set of all pairs $(X,\phi)$ where $X$ is a Riemann surface of genus $g$ and $\phi : S_{g} \...
W.Smith's user avatar
  • 275
1 vote
0 answers
271 views

We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize). Given a surface $S$ we can define Teichmüller space $T(S)$ to ...
Sidharth Ghoshal's user avatar
  • 3,169
5 votes
1 answer
364 views

Given compact Kähler manifolds $X$ and $X'$ deformation equivalent over the unit disk $\Delta \subset \mathbb{C}$. More precisely, there is a proper holomorphic surjective map \begin{align*} \pi\...
David.D's user avatar
  • 423
3 votes
1 answer
450 views

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 ...
giulio bullsaver's user avatar
  • 1,483
4 votes
2 answers
374 views

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 ...
Harry Reed's user avatar
  • 795
7 votes
0 answers
202 views

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 ...
V. Rogov's user avatar
  • 1,420
2 votes
0 answers
121 views

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 ...
Lucien's user avatar
  • 838
3 votes
0 answers
136 views

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 ...
Julien Marché's user avatar
  • 1,351

1
2 3 4 5 6