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### 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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**1**answer

170 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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112 views

### 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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325 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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119 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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575 views

### 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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**2**answers

87 views

### 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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191 views

### 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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283 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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**1**answer

164 views

### 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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249 views

### 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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507 views

### 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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**1**answer

121 views

### 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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105 views

### 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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180 views

### 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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139 views

### 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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69 views

### 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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120 views

### 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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141 views

### 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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141 views

### 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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128 views

### 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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129 views

### 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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190 views

### 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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347 views

### 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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462 views

### 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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235 views

### 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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280 views

### 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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180 views

### 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.
...

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277 views

### Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?

First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...

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### Haken manifolds and characterising sutured manifold hierarchies

In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken ...

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### The measure on the harmonic spectrum from Selberg trace formula

One can see the following two equations,
Theorem 6.1 (Selberg Trace formula) on page 26 of these notes.
Equation 3.19 and 3.20 on page 11 of this paper.
I vaguely feel that these two are the ...

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### Counterexamples to analogue of Cannon conjecture in higher dimensions

It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for ...

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### Fixed points on boundary of hyperbolic group

Let G be a word-hyperbolic group with torsion and let âˆ‚G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on âˆ‚G?

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129 views

### The distance between two farthest points on the Bolza surface?

The Bolza surface $M$ is the closed hyperbolic surface of genus $2$ that can be obtained by identifying the opposite sides of the regular octagon in $\mathbb{H}^2$.
What two points on $M$ are ...

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424 views

### Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$

Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...

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231 views

### Andreev's Theorem and Thurston's hyperbolization theorem

I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, ...

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225 views

### Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...

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152 views

### Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...

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### Existence of special pants decompositions for non-elementary representations into PSL(2,R)

A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation ...

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### Dehn's algorithm for word problem for surface groups

For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...

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137 views

### Discrete Isoperimetric Problem in the Hyperbolic Plane

Let $S_{g}$ be the closed orientable genus $g$ surface. I'm interested in studying a certain class $\Gamma$ of graphs on $S_{g}$ which fill (the complementary regions are simply connected), and which ...

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552 views

### The fundamental group of a closed surface without classification of surfaces?

The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...

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155 views

### Comparing two Delaunay tessellations on a hyperbolic surface

Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb ...

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226 views

### Topology of boundaries of hyperbolic groups

For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was ...

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190 views

### Heegaard genus of covers of cusped hyperbolic 3-manifold

Is there a cusped (non-compact) orientable hyperbolic 3-manifold $M$ which has Heegaard genus $g$, and has a finite-sheeted cover with Heegaard genus $<g$?
Moreover, if the cover is index $n$, then ...

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190 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 ...

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231 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 ...

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394 views

### Distortion of malnormal subgroup of hyperbolic groups

Let $G$ be a countable, Gromov-hyperbolic group.
We say that $H$ is hyperbolically embedded in $G$ if $G$ is relatively hyperbolic to {$H$} (in the strong sense). This definition is due to Osin.
A ...