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1
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1answer
137 views

Some general properties of arithmetic groups of simplest type

I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...
4
votes
1answer
66 views

Geometric realization of an abstract triangulation of the plane

Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy ...
9
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1answer
494 views

Examples of Einstein four-manifolds of negative sectional curvature

Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, ...
1
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0answers
49 views

Eigenvalues to Dirichlet Laplacian in hyperbolic plane

I'm interested in the study of the spectrum of the Laplacian in hyperbolic plane with Dirichlet boundary conditions. My question is whether one knows explicit domains for whom the eigenvalues are ...
8
votes
2answers
239 views

How many knots are there with hyperbolic volume less than a given constant

Are there any known upper bounds on: $$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$ ? I expect this grows at least exponentially in $M$, ...
12
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2answers
643 views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
11
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0answers
101 views

Canonical Immersion of the Double Torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
2
votes
1answer
86 views

Parallel transport on a Hadamard manifold

Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau_{x, w}$ and ...
2
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0answers
155 views

Exotic actions of hyperbolic groups

Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that: The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$. For every quasiconvex subgroup $H \leq ...
5
votes
1answer
528 views

Any way to prove Prime Number Theorem using Hyperbolic Geometry? [closed]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$: $$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$ ...
4
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0answers
78 views

Reference request: 3-dimensional Mobius transforms

I am working on a project that I suspect requires calculations involving Möbius transformations on 3-dimensional space $\mathbb{R}^3$, identified with the quaternions $\mathbb{H}$ with $k$-component ...
8
votes
2answers
1k views

Is there a compass and straightedge construction of parallel lines in hyperbolic geometry?

Is there a compass and straightedge construction of parallel lines in hyperbolic geometry? That is, given a line and a point not on the line, construct a line parallel to the given line.
9
votes
4answers
648 views

Geometry of the space of circles in the Euclidean plane

We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$. It may even be possible to write an explicit formula ...
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0answers
37 views

Is triple point intersection 'generic' in Teichmuller space?

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...
4
votes
1answer
52 views

Additivity of simplicial volume

I have read for example in the introduction of http://arxiv.org/pdf/math/0506338v2.pdf about the property that if we glue hyperbolic manifolds with geodesic boundary consisting of tori along some ...
-1
votes
1answer
57 views

Equal-area projections of the hyperbolic plane [closed]

I'm aware of the projections of the hyperbolic plane at http://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane, but how would I project the hyperbolic plane onto a flat piece ...
4
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0answers
49 views

Bi-Lipschitz classification of germs of conformal metrics at a singularity

First let me introduce some definitions. By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in ...
4
votes
2answers
708 views

Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length. Of course, the ...
5
votes
1answer
182 views

Uniformization of a plane minus cantor set

Let $\mathbb{D}$ be the unit disk endowed with the Poincaré metric and $G$ be a Fuchsian group such that the hyperbolic surface $\mathbb{D}/G$ is homeomorphic to the plane minus a Cantor set. ...
0
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1answer
101 views

the paraboloid model for hyperbolic space [closed]

In Thurston's Three-Dimensional Geometry and Topology, he gives a recipe for a non-standard model for hyperbolic space which he calls the paraboloid model. I'd like to use the model to try out ...
13
votes
2answers
391 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 ...
14
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3answers
476 views

Failure of Mostow rigidity in dim. 2

I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question: (1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ ...
18
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2answers
510 views

Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...
1
vote
1answer
80 views

What is the expression of first eigen function of Laplacian on Hyperbolic plane?

Let $\Delta$ be the Laplacian (a positive operator) on $H^2$ the hyperbolic plane. My question is what is the expression of the eigenfunction $\Delta f= f/4$? (say in the ploar coordiante)
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6answers
1k views

Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact: Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...
1
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0answers
73 views

Hlawka inequality for Lorentz quadratic form

Let $K$ be a convex cone in ${\mathbb R}^n$. A continuous function $f:K\rightarrow\mathbb R$ satisfies a Hlawka inequality if $$f(0)+f(x+y)+f(y+z)+f(z+x)\le f(x)+f(y)+f(z)+f(x+y+z),\qquad\forall ...
3
votes
1answer
156 views

δ-hyperbolic space

It is known that A δ-hyperbolic space is a geodesic metric space in which every geodesic triangle is δ-thin. (http://en.wikipedia.org/wiki/%CE%94-hyperbolic_space) The question is that if we remove ...
7
votes
1answer
241 views

Does the limit in the Volume conjecture converge?

The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then $$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$ ...
4
votes
1answer
124 views

Angle between geodesics in hyperbolic surface

Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma_1$ and $\gamma_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic ...
3
votes
1answer
85 views

simplex in hyperbolic space and quadrics in projective space

I am trying to understand Goncharov's paper "Volumes of hyperbolic manifolds and mixed Tate motives", http://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00293-3/S0894-0347-99-00293-3.pdf In ...
5
votes
2answers
397 views

Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...
0
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2answers
113 views

Metric properties of a quadratic differential at an essential singularity

Let $f(z)dz^2$ be a holomorphic quadratic differential on the punctured disk $\{0<|z|<1\}$, which gives rise to a Riemannian metric $g=|f(z)|\,|dz|^2$ and hence a volume form $\nu=|f(z)| ...
3
votes
1answer
215 views

Dirichlet polyhedra for hyperbolic manifolds

Let $H$ be a simply-connected, complete space of constant negative curvature, that is, a hyperbolic space, $\Gamma$ a discrete group of isometries, and and $M=H/\Gamma$ its quotient space; we assume ...
2
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1answer
106 views

Intersection of closed geodesics in hyperbolic surface

This question may be easy but I could not come up with a proof. Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed ...
6
votes
2answers
179 views

Geodesic flow on infinite surfaces

The geodesic flow on a compact hyperbolic surface (i.e. a surface with a riemannian metric of constant curvature $-1$) has been well-studied, in particular it has been known for a long time that it is ...
7
votes
1answer
266 views

For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?

The fractal dimension of the 3D Apollonian packing is computed in this paper. In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension ...
2
votes
0answers
58 views

Are Zariski-dense representations of a cocompact complex hyperbolic lattice non-obstructed?

Question Suppose that $\Gamma < \text{SU}(n,1)$ is a cocompact lattice, and let $\rho \colon \Gamma \to G$ be a representation to a non-compact simple Lie group (most interesting case for me: $G = ...
2
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1answer
174 views

Can the finiteness of a Burnside group with two generators be checked algorithmically by using Fuchsian von Dyck groups?

BIG EDIT of the previous question "Coverings of the free Burnside groups", never answered. In the paper http://link.springer.com/article/10.1007/BF00046586 (last section) there is an interesting ...
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0answers
88 views

Finding Riemannian metric explicitly from the conformal structure on a surface

Each Riemann surface structure $S'$ on a topological surface $S$ can be obtained from a Riemannian metric which looks like $ds^{2} = g_{11}dx^{2} + 2g_{12}dxdy + g_{22}dy^{2}$ in local coordinates ...
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6answers
2k views

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 ...
8
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1answer
482 views

Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?

In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...
10
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2answers
308 views

Heegaard genera of arithmetic 3-manifolds

UPDATE: Because I was hoping that state the question as concisely as possible, the original post did not include a precise definition of arithmetic 3-manifold only a reference to Maclachlan and ...
9
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0answers
414 views

How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $G/K$ is a symmetric space of noncompact type, $g=k+p$ the ...
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0answers
73 views

Bloch group, hyperbolic manifolds and rigidity

I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to ...
1
vote
1answer
79 views

Reference for the result that the systol map from Teichmuller space to curve complex is coarsely Lipschitz

Let $\mathcal{T}(S)$ denotes the Teichmuller space of a finite type surface $S$ equipped with Teichmuller metric and $\mathcal{C}(S)$ denotes the curve complex. Define a map ...
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1answer
293 views

Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
4
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2answers
169 views

Visibility spaces and Gromov hyperbolicity

I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic ...
1
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1answer
130 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 ...
1
vote
2answers
109 views

Does the Teichmüller space of the pair of pants admit a continuous global section?

Let $P$ be a pair of pants, $H(P)$ be the space of smooth hyperbolic Riemannian metrics with geodesic boundary on $P$, and $T(P)$ be the Teichmüller space of $P$ (quotient of $H(P)$ under smooth ...
2
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1answer
117 views

Why do noncocompact arithmetic Kleinian groups have quadratic trace fields?

I realize there are a few different ways of going about proving this, depending on one's background, but there's a particular number theoretic aspect that I am just blanking on, and can't seem to find ...