Skip to main content

All Questions

Filter by
Sorted by
Tagged with
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
1 answer
660 views

On trivial mapping class group of 3-manifolds

What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
Anubhav Mukherjee's user avatar
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
7 votes
0 answers
218 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
6 votes
1 answer
166 views

Translation length on annular curve graphs

Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked. Definition: let the group $G$ act by isometries on a metric space $(X,d)$. ...
Mark Hagen's user avatar
5 votes
1 answer
333 views

Proof of homotopic essential simple close curves are isotopic

In the book by Benson Farb and Dan Margalit A primer on mapping class groups, Princeton Mathematical Series 49. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/...
T566y65tt's user avatar
  • 119
5 votes
1 answer
204 views

Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface

If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups: $$\text{Mod}(\mathcal{R}')\longrightarrow \...
QGravity's user avatar
  • 989
4 votes
1 answer
463 views

Hyperbolic three-manifolds that fiber over the circle

Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...
user524868's user avatar
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
4 votes
1 answer
609 views

About isotopy and homotopy

In the " A Primer on Mapping Class Groups Benson Farb and Dan Margalit" We have : Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\...
T566y65tt's user avatar
  • 119
4 votes
1 answer
195 views

Ideal triangulation of hyperbolic 3-manifold with generic mapping class group

I am from physics background so I apologize in advance if my question is trivial. Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$...
Zhengdi Sun's user avatar
3 votes
1 answer
187 views

2-orbifolds that I expect to be hyperbolic, but they're nonnegatively curved

I'm considering some complex 1-dimensional/real 2-dimensional orbifolds that I expect to be hyperbolic. However, some of them seem to be Euclidean or spherical. Any thoughts what's going on here? Here ...
Ethan Dlugie's user avatar
  • 1,277
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
2 votes
2 answers
197 views

Can a hyperbolic three-manifold have 𝑛 toric boundary components?

I'm wondering if I can construct a class of hyperbolic three-manifolds with $n$ toric boundaries. My idea is to take a bordered hyperbolic Riemann surface $\Sigma_{g,n}$, of genus $g$ with geodesic ...
Oblonski's user avatar
  • 133
2 votes
1 answer
136 views

Example of maximal multicurve complex

in this paper we have : " On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps." Definition. The maximal multicurve complex $...
Usa's user avatar
  • 119
1 vote
0 answers
153 views

Topological entropy and pseudo-Anosov dilatation for punctured surface

Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ punctures. Assume $2-2g-n<0$. Let $f$ be a pseudo-Anosov mapping class with dilatation $\lambda_f$. In the introduction (1st page) of the ...
Cusp's user avatar
  • 1,713
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