All Questions
25 questions
5
votes
1
answer
104
views
When do two measured foliations on a surface define a Riemann surface structure?
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,\...
- 275
5
votes
1
answer
447
views
Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$
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 ...
- 153
0
votes
1
answer
133
views
Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds
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{...
- 337
4
votes
2
answers
261
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 ...
- 765
1
vote
0
answers
397
views
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 ...
user318929
6
votes
2
answers
302
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 ...
- 61
5
votes
1
answer
248
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 ...
- 1,435
3
votes
1
answer
853
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 ...
- 1,435
1
vote
0
answers
218
views
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 ...
- 21
13
votes
2
answers
485
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 ...
- 10.1k
3
votes
0
answers
196
views
Ending lamination theorem
Let $M$ a compact manifold with surfaces $S_1,...,S_p$ as boundaries. Let us suppose that $M$ admits a complete hyperbolic structure. Then, from the ending lamination theorem, given either laminations ...
user50806
9
votes
2
answers
385
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 ...
- 2,390
0
votes
2
answers
219
views
If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?
Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...
- 3,751
3
votes
0
answers
414
views
Geometric intersection number for product of elements of the fundamental group
Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...
- 1,713
8
votes
2
answers
631
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 ...
- 81
8
votes
2
answers
566
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 $...
- 1,713
1
vote
0
answers
143
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 ...
- 1,713
8
votes
1
answer
601
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 ...
- 81
3
votes
0
answers
198
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 ...
- 31
9
votes
2
answers
233
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 ...
- 540
1
vote
1
answer
271
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 ...
- 829
7
votes
2
answers
335
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 ...
- 540
4
votes
1
answer
1k
views
Hyperbolic structures on once punctured tori
I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.
My ...
- 1,601
14
votes
2
answers
617
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 ...
- 540
0
votes
2
answers
863
views
A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem
Hello,
I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...
- 1,471