Questions tagged [teichmuller-theory]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
49 votes
4 answers
22k views

Consequences of Kirti Joshi's new preprint about p-adic Teichmüller theory on the validity of IUT and on the ABC conjecture

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 ...
Madeleine Birchfield's user avatar
40 votes
1 answer
18k views

What is "Teichmüller Theory" and its history?

What is "Teichmüller Theory"? What part has been worked out / foreseen by O. Teichmüller himself and what is further development? Is there some current work which might be considered as continuation/...
Alexander Chervov's user avatar
24 votes
2 answers
2k views

Explicit homeomorphism between Thurston's compactification of Teichmuller space and the closed disc

Thurston's celebrated compactification of Teichmuller space was first described in his famous Bulletin paper. Teichmuller space is famously homeomorphic to an open disc of some dimension (this can be ...
Bruno Martelli's user avatar
20 votes
7 answers
7k views

What are some Applications of Teichmüller Theory?

I'm trying to collect some specific examples of applications of Teichmüller Theory. Here are some things I have collected thus far: No-wandering-domain Theorem (Sullivan) Theorems of Thurston (...
20 votes
1 answer
2k 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 ...
user avatar
18 votes
1 answer
1k views

The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions

I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...
user331406's user avatar
15 votes
2 answers
4k views

Switching from pure mathematics (e.g. geometry) to more applied areas (e.g imaging) after Ph.D., as postdoc and chance of getting such a postdoc?

Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point ...
15 votes
2 answers
824 views

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 ...
Jacques's user avatar
  • 563
15 votes
0 answers
1k views

reference - Grothendieck on Thurston's work

In his 'dernières' years Grothendieck gets "interested" in Thurston's work. "[...] je me suis intéressé ces dernières années - la géométrie hyperbolique à la Thurston et ses relations au groupe de ...
tttbase's user avatar
  • 1,700
14 votes
3 answers
598 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 $...
Charlie Frohman's user avatar
14 votes
2 answers
602 views

For which surfaces is Penner's conjecture known to be true?

Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along ...
leone slavich's user avatar
14 votes
1 answer
732 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....
Lyle Ramshaw's user avatar
13 votes
3 answers
1k views

Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates

The Teichmüller space $T_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}_+^{3g-3}$. The ...
Yuji Tachikawa's user avatar
13 votes
2 answers
471 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 ...
Dylan Thurston's user avatar
12 votes
2 answers
1k views

Universal covering of a 2-sphere without $n$ points

Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic ...
asv's user avatar
  • 21.1k
10 votes
3 answers
883 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 ...
Kim's user avatar
  • 4,034
10 votes
4 answers
1k views

Analytic function avoiding elements of the modular group

A friend recently told me the following two facts, for which he cannot recall a proof or a reference (but he remembers seeing them in the literature): Let $f$ be a holomorphic function mapping the ...
Alexandre Eremenko's user avatar
9 votes
2 answers
3k views

What is / are the softwares to use to draw surfaces of the form of a two or three-holed torus , or torus, or torus with cusps attached to it?

I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with ...
Analysis Now's user avatar
  • 1,451
9 votes
2 answers
230 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 ...
leone slavich's user avatar
9 votes
2 answers
276 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 ...
user470881's user avatar
9 votes
2 answers
358 views

Are pseudo-Anosov foliations dense?

A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of ...
Adam's user avatar
  • 2,370
9 votes
1 answer
850 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 ...
Vesselin Dimitrov's user avatar
9 votes
1 answer
164 views

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 $\Sigma_g$...
Selim G's user avatar
  • 2,626
9 votes
0 answers
166 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 ...
F. Germano's user avatar
8 votes
2 answers
533 views

Pseudo-Anosov maps with same dilatation.

Let $S$ be a hyperbolic surface. Suppose $\mathcal{T}$ denotes the Teichmuller space of $S$ and $Mod(S)$ denotes the mapping class group of $S$. Given any pseudo-Anosov element $f\in Mod(S)$, suppose $...
Cusp's user avatar
  • 1,703
8 votes
1 answer
368 views

Is Teichmüller distance bigger than Weil-Petersson distance on Teichmüller space?

It is known that Teichmüller distance ($d_{Teich}$) on Teichmüller space is complete, whereas Weil-Petersson distance ($d_{WP}$) is not complete. See for example the article Wolpert, Scott. ...
user197284's user avatar
8 votes
1 answer
384 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,...
Dylan Thurston's user avatar
8 votes
2 answers
571 views

Teichmüller space on non-orientable closed surfaces

It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a ...
Curious's user avatar
  • 81
8 votes
1 answer
570 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 ...
Redin's user avatar
  • 81
7 votes
1 answer
5k views

Why use Teichmuller representatives?

In p-adic mathematics, what is the advantage of using Teichmuller representatives over using just the numbers 0,1,2,...,p-1 ? In either case, the norm is the same. In either case, all the points are ...
Robert Paster's user avatar
7 votes
1 answer
818 views

Teichmuller groupoids in Grothendieck's esquisse d'un programme

Grothendieck in his Esquisse d'un programme mentioned without any precise definition and construction that the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on the whole "tower" ...
asv's user avatar
  • 21.1k
7 votes
2 answers
520 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 boundary ...
Anonymous's user avatar
  • 818
7 votes
1 answer
807 views

Why is the length spectrum called a spectrum?

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
7 votes
2 answers
887 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 ...
Cusp's user avatar
  • 1,703
7 votes
2 answers
328 views

Comparing different layered structures for fibered 3-manifolds: example request.

Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular ...
leone slavich's user avatar
7 votes
2 answers
282 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 ...
Maxime Scott's user avatar
7 votes
3 answers
319 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 ...
Lisa's user avatar
  • 71
7 votes
1 answer
378 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 ...
aglearner's user avatar
  • 14k
7 votes
1 answer
365 views

Selberg Zeta Function and Fenchel-Nielsen Coordinates

According to Uniformization theorem every compact Riemann surface $\Sigma$ of genus $g\ge2$ is isomorphic to a space that can be obtained by the action of a Fuchsian group on upper half plane $\mathbb{...
QGravity's user avatar
  • 969
7 votes
1 answer
371 views
+200

Goldman symplectic form vs Weil–Petersson symplectic form

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
  • 38
7 votes
0 answers
154 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 ...
 V. Rogov's user avatar
  • 1,115
7 votes
0 answers
325 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} ...
EtienneBfx's user avatar
7 votes
0 answers
198 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 ...
Mauro's user avatar
  • 191
7 votes
0 answers
377 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 ...
user avatar
6 votes
4 answers
672 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 ...
Kim's user avatar
  • 4,034
6 votes
2 answers
464 views

Riemann Theta Function On Hyperbolic Riemann Surfaces

The Riemann theta function for a genus $g$ closed Riemann surface with period matrix $\tau=[\tau_{ij}]$ is defined by $$\theta(\{z_1,\cdots,z_g\}|\tau)=\Sigma_{n\in\mathbb{Z}^g}e^{\pi i(n\cdot\tau\...
QGravity's user avatar
  • 969
6 votes
1 answer
377 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 ...
Anonymous's user avatar
  • 818
6 votes
2 answers
278 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 ...
Tina's user avatar
  • 61
6 votes
1 answer
747 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 ...
Anonymous's user avatar
  • 818
6 votes
1 answer
182 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 ...
user avatar

1
2 3 4 5 6