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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 ...
leone slavich's user avatar
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 ...
Dylan Thurston's user avatar
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 ...
leone slavich's user avatar
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 ...
Adam's user avatar
  • 2,390
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 $...
Cusp's user avatar
  • 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 ...
Curious's user avatar
  • 81
8 votes
1 answer
600 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
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 ...
leone slavich's user avatar
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 ...
Tina's user avatar
  • 61
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,\...
W.Smith's user avatar
  • 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 ...
Chaitanya Tappu's user avatar
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 ...
giulio bullsaver's user avatar
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 ...
b b's user avatar
  • 1,601
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 ...
Harry Reed's user avatar
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 ...
giulio bullsaver's user avatar
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 ...
user avatar
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 ...
Cusp's user avatar
  • 1,713
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 ...
user3419's user avatar
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 ...
user126154's user avatar
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 ...
user avatar
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 ...
user117580's user avatar
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 ...
Cusp's user avatar
  • 1,713
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, ...
Analysis Now's user avatar
  • 1,471
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{...
JMK's user avatar
  • 337
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 ...
Anubhav Mukherjee's user avatar