The hyperbolic-geometry tag has no wiki summary.

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### Generalizations of Belyi's theorem

Belyi's theorem states that the following properties of a nonsingular projective algebraic curve $X$ are equivalent:
1) $X$ is defined over $\overline{\mathbb{Q}};$
2) There exists a meromorphic ...

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### A question about the convention for the Plancherel measure on $\mathbb{H}^n$

Say I have to calculate the quantity, $Log Tr [ -\Delta - \frac{1}{4} + m^2]$ on $H^n$. Then looking up the spectral measure $\mu(\lambda)$ and the eigenvalues of the Laplacian ($= -\Delta = - ...

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### Laurent series of an integral over the Plancherel measure over hyperbolic space

So for $n \in \mathbb{Z}, m \in \mathbb{R}, q>0$ I am looking at this integral,
$\int_0 ^\infty dx \frac{\Gamma(ix + \frac{1+y}{2}) \Gamma(-ix + \frac{1+y}{2}) }{\Gamma(ix) \Gamma(-ix) } \left [ ...

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280 views

### Straight line on the Poincare disk hitting points almost everywhere

Consider the tiling of the Poincare disk $\mathbb{D}$ by identified octagons (i.e., representing a torus with genus 2). Suppose inside each octagon is a subset A such that the octagon minus A is a ...

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222 views

### Two questions on isometric embedding

According to the answer of the following question, I try a new version:
An special isometric embedding
Let $M$ be a Riemannian manifold and $\gamma$ be a small part of a geodesic.
Is there an ...

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120 views

### Phillips-Sarnak conjecture in higher dimension

The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...

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167 views

### What's the height of the capped hyperbolic pants?

Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_1,l_2,l_3$. Cap off boundary 1 with a conformal disk. The result is a conformal cylinder, which has a unique flat ...

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### Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...

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**1**answer

117 views

### Change of coordinates for Teichmüller space of the 4-holed sphere

The diagram below indicates 2 ways to use Fenchel-Nielsen coordinates to parameterize the Teichmüller space of conformal structures on the 4-holed sphere with totally-geodesic boundary, corresponding ...

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### Distances between boundaries in a hyperbolic pants

Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_i$ for $i \in \{1,2,3\}$. For any two distinct boundary components, is the length of the shortest geodesic connecting ...

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### Reference request: embedding the hyperbolic triangulation in $\mathbb{R}^3$

Let $T_d$ be the infinite valence $d$ triangulation of the hyperbolic plane, where each triangle is equilateral and $d \ge 7$. Question: Is there an isometric embedding from $T_d \to \mathbb{R}^3$? ...

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### How much do the classes of geodesic metrics and Green metrics on a Gromov hyperbolic group differ?

Background
Let $G$ be a Gromov hyperbolic group. If $G$ acts properly discontinuously and cocompactly on a proper geodesic metric space $X$, then any bijective identification of $G$ with its orbit ...

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**1**answer

118 views

### Entropy of Negatively pinched manifolds

Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. ...

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### Virtual fibering conjecture for cusped hyperbolic manifolds

I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case.
Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the ...

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176 views

### Rank and hyperbolic volume

Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of ...

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### Mid point with set square?

Is it possible to construct the midpoint of a segment in the hyperbolic plane
using the set square only?
With the set square one can
draw the line through the given two points and
drop the ...

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241 views

### Topological rigidity for negatively curved manifolds?

I was wondering if two compact oriented manifold carrying a Riemannian metric with negative sectional curvature, whose fundamental groups are isomorphic, are necessarily diffeomorphic (or ...

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389 views

### When are those subgroups of $\mathrm{SL}(2, \mathbb{C})$ discrete?

Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter.
Now consider the family of representations ...

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263 views

### Who first used the cross-ratio to describe shapes in hyperbolic geometry?

I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes
and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic ...

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### Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question:
"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically
immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"
I believe the answer to ...

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152 views

### Model of hyperbolic geometry with finite number of parallel line

Does there exist a model of hyperbolic geometry such that only finite number of distinct parallel lines through a point which does not intersect given line?
Edit (Misha): I usually do not edit other ...

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284 views

### What constant ensures hyperbolicity of Dehn surgery?

I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...

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### Some mid-sized ¿hyperbolic? manifolds and SnapPea

I've recently come across some mid-sized 3-manifolds that I think are likely hyperbolic, but SnapPea has some trouble with them. This is related to my previous question
can you fool SnapPea?
but in ...

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### Is there a criterion for a link complement to have a hyperbolic structure with finite volume

For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with ...

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106 views

### About a definition of quasi-conformal maps

A book I'm reading gives the following definition for quasi-conformal maps:
If $f$ is a homeomorphism of a metric space X to itself, $f$ is K-quasi-conformal if and only if for all $z \in X$:
...

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### Is there a volume conjecture for closed 3-manifolds?

A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is
Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...

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### Subharmonic function on a twice punctured complex plane

is the twice punctured complex plane parabolic or hyperbolic? In this sense: does $\mathbb{C}-\{0,1\}$ admit a nonconstant, negative, subharmonic function?
Thanks,

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### Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic n-space. Let $TH$ be the tangent bundle of $H$, endowed with its Sasaki metric. I have two questions:
Is $TH$ isometric to $H$ times a flat n-space?
What is the group of ...

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### hyperbolic orbifolds of small area

Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller ...

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### What is known about maximal free subgroups of surface groups?

Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...

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### Geodesics on the twisted pseudosphere (Dini's surface)

I wonder how difficult it is to compute geodesics on Dini's Surface,
a twisted pseudosphere?
Here is one parametrization, from
Alfred Gray's Modern Differential Geometry of Curves and Surfaces, ...

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### Density of ends of long words in a hyperbolic group

Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...

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### Fixed directions and Zariski density of hyperbolic groups

It is a fact that if $\Lambda$ is a nonelementary subgroup of ${\rm PSL_2}(\mathbb{C})$ which contains an hyperbolic transformation and moreover ${\rm tr}(g)\in\mathbb{R}/\pm 1$ for all $g\in\Lambda$ ...

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### Is $\mathbb{H}^n$ quasi-isometric to a leaf of a codimension 1 foliation of a compact manifold?

If we extend the action of $\pi_1(\Sigma_g), g\geq 2,$ from $\mathbb{H}^2$ to its boundary $\partial_{\infty}\mathbb{H}^2=S^1$, the surface bundle corresponding to this action of $\pi_1(\Sigma_g)$ on ...

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### Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?

Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order. ...

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### Discussion of specific arithmetic triangle groups?

Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ...

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### Classification of maximal nonuniform Fuchsian lattices existent?

I am interested in the set of all non-cocompact Fuchsian lattices which all have a distinguished point as cusp, say $\infty$ in the upper half plane model of the hyperbolic plane. Of course, the ...

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### Origin of spectral theory on infinite-area hyperbolic surfaces

The study of spectral theory of finite-area hyperbolic surfaces is intimately related to number theory, in particular by the importance of Maass cusp forms. The counting of resonances is of ...

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### Eigenvalues vs resonances

I understand that for infinite-area hyperbolic surfaces, there are no $L^2$-eigenfunctions of the Laplace-Beltrami operator but there are a lot resonances.
But I am confused about the notion of ...

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### How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...

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### Fundamental group of an hyperbolic $4$-manifold

Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide ...

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### Euclidean surfaces with conical singularities and cusped hyperbolic surfaces

Let $S$ be a compact orientable surface endowed with a singular euclidean metric $g$, with $n$ conical singularities $x_1,\ldots,x_n$.
Construction 1: it is well-known that the conformal class ...

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### Hyperbolic Manifolds which fiber over the circle

If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...

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### Injective simplicial maps between Arc complexes

Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...

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### Right-angled polytopes

%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not ...

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### The number of cusps of higher-dimensional hyperbolic manifolds

Suppose $n$ is an integer greater than 3. Sometimes ago I heard somewhere that it is still not known if there exist complete finite-volume hyperbolic $n$-manifolds having exactly one cusp.
Could ...

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### Geometrization & JSJ decomposition with boundary

Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...

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### Do elements of the fundamental group give rise to isometries

Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...

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### How close can closed geodesics be?

A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such ...

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### A question on Cayley graphs and hyperbolic 3-manifolds

There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...