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-1
votes
0answers
70 views

Embedded Minimal Surfaces in 3D Hyperbolic Space [closed]

My question concerns the embedding of minimal surfaces in the 3D hyperbolic space. The minimal surface is defined in terms of Weierstrass-Enneper representation. Let us take now a slice (section) of ...
1
vote
0answers
101 views

What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp

First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...
1
vote
0answers
46 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 ...
4
votes
1answer
91 views

Non-Haken hyperbolic 3-manifolds without nonorientable surfaces

It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. ...
2
votes
0answers
45 views

Estimates for hyperbolic metrics on nested surfaces

Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} ...
4
votes
0answers
149 views

Computing algebraic properties of trace fields, as given by SnapPy

SnapPy can tell you the trace field of a hyperbolic $3$-manifold (which is awesome), but it specifies the field by outputting: the minimal polynomial of the field over $\mathbb{Q}$, and a decimal ...
30
votes
4answers
1k views

Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
7
votes
0answers
79 views

Compatibility of spherical and hyperbolic geometry for fibred knots

Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
1
vote
0answers
100 views

Extending continuous functions from $\partial X$ to $X\cup \partial X$

Consider a proper geodesic hyperbolic space $X$ (in the sense of Gromov). Let $\partial X$ be its Gromov boundary. Consider a complex-valued continuous function on the boundary $f\colon\partial ...
4
votes
1answer
160 views

How many quadratic fields occur as trace fields of hyperbolic knot complements?

I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs ...
4
votes
0answers
139 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 ...
6
votes
3answers
255 views

For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field?

Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, ...
3
votes
1answer
171 views

How many non-commensurable non-arithmetic manifolds have a quaternion algebra like this?

I am interested in realizing commensurability classes of hyperbolic $3$-manifolds whose quaternion algebra (note: not invariant quaternion algebra) is isomorphic to one of the form ...
3
votes
1answer
98 views

The action of an S-arithmetic group on the hyperbolic plane

I have a really quick question. I am interested in $G=SL_2(\mathbb{Z}[1/p_1,...,1/p_n])$, where $p_1$,..., $p_n$ are prime numbers. Since $G$ is a subgroup of $SL_2(\mathbb{R})$, it acts in the ...
1
vote
0answers
60 views

Comparison theorem for Lambert quadrilateral

A Lambert quadrilateral is a quadrilateral three of whose angles are right angles. And in 2-d hyperbolic space $\mathbb H^2$, we have nice formulas for the fourth angle. If $AOBF$ is a Lambert ...
2
votes
0answers
47 views

Uniform convergence of long geodesic to Liouville measure

Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...
10
votes
3answers
296 views

Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture: A non-trivial connected sum $M_1\# M_2$ admits a ...
3
votes
0answers
115 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 ...
0
votes
1answer
57 views

Geodesics under deck transformations on hyperbolic surface

I have a question regrading the deck transformations. As we know, for the 2-torus $\mathbb T^2$, if we have a geodesic $\widetilde\gamma$ on the corresponding covering space $\mathbb R^2$, and ...
4
votes
1answer
105 views

Uniqueness of hyperbolic rescaling

Let $X$ be a compact oriented surface of genus at least two, equipped with a Riemannian metric $g$. By the uniformization theorem for Riemann surfaces, there is a conformal universal covering map ...
6
votes
0answers
142 views

Is there any exotic smooth structure on open hyperbolic manifold?

I edited my post to clarify some confusions as suggested by Igor. Let $M$ be an open hyperbolic manifold, with or without finite volume, Is there any manifold $N$ which is homeomorphic to $M$ but ...
5
votes
1answer
339 views

Relations between some works by Deligne-Mostow and Thurston

Happy new year 2016! A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. ...
3
votes
1answer
189 views

Kobayashi distance function on the upper half-space

I asked this question already in mathstackexchange but got no answer, so I ask it again here. Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ ...
3
votes
1answer
102 views

What does the trace of a loxodromic Mobius transformation tell us about how it rotates?

A matrix $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{PSL}_2(\mathbb{C})$ acts isometrically on the upper half-space model $\mathbb{H}^3$ via isometric extension of the Mobius ...
7
votes
3answers
204 views

In the $\mathbb{H}^3$ upper half space model, is a hemiellipsoid perpendicular to the plane at infinity a minimal surface?

Given a Jordan curve on the $\mathbb{H}^3$ boundary at infinity, there is a minimal surface (topological disk) in $\mathbb{H}^3$ with the curve as its asymptotic boundary ...
2
votes
1answer
138 views

Blaschke Condition for hyperbolic lattices

For $r$, $s$, small positive integers, do the complex numbers on the unit disc (without the hyperbolic metric) corresponding to the vertices of the hyperbolic tiling with Schläfli symbol $\{r,s\}$ ...
9
votes
2answers
299 views

Volume form on a hyperbolic manifold with geodesic boundary

Let $M$ be a compact connected orientable Riemannian $n$-manifold with boundary $\partial M\ne\emptyset$. Since $H^n(M,\mathbb R)=0$, the connecting morphism $\delta: H^{n-1}(\partial M,\mathbb R)\to ...
0
votes
0answers
81 views

Hyperbolic Space, Lattice Isometries, and Polyhedral Fundamental Domains

Let $L$ be a lattice of signature $(1,n)$. Suppose I have a (probably infinite index) subgroup $\Gamma\subset O^+(L)$ of the isometries of $L$ which preserve the positive cone $\mathcal{C}^+\subset ...
3
votes
2answers
328 views

Subgroups of hyperbolic groups

Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated? In view of Ian Agol's answer, I ...
4
votes
1answer
196 views

Compact open topology on the space of geodesics

I'm new in the field, so I'm sorry in advance if my question is too naive. Let's consider $S$ a surface of genus $g\ge 2$ with an hyperbolic metric $g$. Let's call $\mathcal{S}(S)$ the set of closed ...
1
vote
1answer
92 views

Lamination as limit of arcs

I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
6
votes
2answers
215 views

How bad is the modular space?

I'm wondering if there is some results about the quotient space $\mathbb{H}^{3}/PSL(2,\mathcal{O}_{K})$? Do we know something about its homology or homotopy groups ? $\mathbb{H}^{3}$ is the hyperbolic ...
24
votes
3answers
811 views

Intuition for Zagier's theorem for $\zeta_K(2)$

In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$: $$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v ...
2
votes
1answer
101 views

Why simple closed curves are dense in $\mathcal{PML}_0(S)$?

I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}_0(S)$ be the set of projective ...
3
votes
1answer
121 views

Why is $\mathcal{PML}_0(S)$ compact?

I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}_0(S)$, the space of projective classes of measured geodesic laminations ...
4
votes
1answer
223 views

The hyperbolic metric on a flat surface

Let $S$ be a closed oriented surface of genus $\geq 2$ and $\mathcal{F}_n(S)$ be the space of flat metrics with conic singularities on $S$ whose cone angles are of the form $2k\pi/n$ ...
9
votes
3answers
387 views

Isometry group of a compact hyperbolic surface

Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have ...
4
votes
0answers
61 views

Is there a formula for the A-model partition function in terms of hyperbolic structure?

The A-model partition function is a topological invariant of any hyperbolic surface. So, what is this invariant? I'm not sure if there is a way to normalize these since the usual perscription $Z(S^2) ...
1
vote
0answers
73 views

Algorithm to generate hyperbolic metric on a compact surface

Let $F$ be a compact surface of genus $g$ with generators $a_1,b_1,\ldots ,a_g,b_g$ with relation $[a_1,b_1]\cdots [a_g,b_g]=1$. We can also consider surfaces with boundary in which case the ...
3
votes
0answers
47 views

Length and laplacian spectrum for quasi-fuchsian manifold

It is well known that, in the case of finite area hyperbolic surfaces, the length sprectrum (the collection of length of all closed geodesics) and the spectrum of the laplacian (acting on functions) ...
5
votes
0answers
134 views

A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers

I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that a) the graph $\Gamma$ is fine, b) $\Gamma$ is not a tree, c) not all ...
4
votes
1answer
131 views

Is the heat kernel for the hyperbolic plane uniformly continuous in $t\in(0,\infty)$?

let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel. I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u_t(x):=k(t,x,y)$ is continuous ...
7
votes
1answer
166 views

Conditions on the hierarchy for Thurston's hyperbolization theorem

From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, ...
7
votes
2answers
209 views

Heegard genus of hyperbolic Haken 3-manifolds

Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?
5
votes
1answer
253 views

How can we explicitly verify a canonical Dirichlet domain for this hyperbolic punctured torus

This is a follow-up question to a question from math.stackexchange: http://math.stackexchange.com/q/1436253/67563 Had it not been for the exchange there between myself and @Lee_Mosher in the comments ...
22
votes
2answers
663 views

SnapPea for the uninitiated

SnapPea (http://www.math.uic.edu/~t3m/SnapPy/) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the ...
2
votes
0answers
91 views

What is the metric on the Fuchsian model? [closed]

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. How is the distance between any two points $x, y \in \mathbb{H} / \Gamma$ in the Fuchsian model ...
5
votes
0answers
86 views

Maximum relator and hyperbolicity

It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows: Suppose ...
4
votes
1answer
133 views

local quasi geodesics in hyperbolic spaces

I asked this question on math stackexchange (see here) but didn't get any answer so I thought I post it here too. We have the following two well-known Theorems: T1) For all $\delta > 0, ...
2
votes
1answer
40 views

Is the length function associated with the twist parameter an increasing function?

Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is ...