The hyperbolic-geometry tag has no usage guidance.

**16**

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430 views

### Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...

**11**

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**0**answers

178 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 ...

**10**

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224 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 ...

**10**

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**0**answers

473 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 ...

**9**

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**0**answers

376 views

### Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
...

**8**

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**0**answers

194 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 ...

**8**

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**0**answers

236 views

### Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$

Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...

**7**

votes

**0**answers

110 views

### Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?

"Universal" means that the construction steps are independent of the length of the given segment. In the Euclidean plane one can take the diagonal of a square built on it. Without the "universal" the ...

**7**

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**0**answers

97 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 ...

**6**

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**0**answers

133 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 ...

**6**

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**0**answers

144 views

### 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 ...

**6**

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**0**answers

177 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 ...

**6**

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**0**answers

317 views

### A conjecture of Thurston and possibly Weeks too

What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...

**6**

votes

**0**answers

560 views

### Closed geodesics on a closed, negatively curved Riemannian manifold

I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...

**6**

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**0**answers

221 views

### Does every hyperbolic 3 manifold with totally geodesic boundary has some finite covering space with more than one boundary component?

I am thinking about the question that: if we double a hyperbolic 3 manifold along its boundary, will the rank of fundemental group of the resulting closed manifold be strictly larger than before?\
The ...

**5**

votes

**0**answers

117 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 ...

**5**

votes

**0**answers

82 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 ...

**5**

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**0**answers

245 views

### When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

**5**

votes

**0**answers

99 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 ...

**4**

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**0**answers

57 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) ...

**4**

votes

**0**answers

64 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

**0**answers

161 views

### 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 ...

**3**

votes

**0**answers

92 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 ...

**3**

votes

**0**answers

44 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) ...

**3**

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**0**answers

80 views

### Barycentric interpolation in hyperbolic triangles

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...

**3**

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**0**answers

137 views

### $\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$.
Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...

**3**

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**0**answers

80 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 ...

**3**

votes

**0**answers

149 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 ...

**3**

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**0**answers

145 views

### reference request: “p-adic” presentation of surfaces

On several occasions I heart about the following result:
For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...

**3**

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**0**answers

308 views

### Boundary defining functions for hyperbolic surfaces

Let $M$ be a geometrically finite hyperbolic surface with one cuspidal end and one funnel end so that it can be divided into $C \cup K \cup F$ where $C$ is the cusp, $F$ the funnel and $K$ the ...

**3**

votes

**0**answers

160 views

### Collapsing the medial axis of a polytope

Let X be a convex polyhedron in hyperbolic 3-space.
Let M be the medial axis of X.
Question: Is M collapsible?
It is easy to see that M is contractable.
In the case of Euclidian 3-space, instead ...

**2**

votes

**0**answers

171 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 ...

**2**

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**0**answers

70 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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**0**answers

58 views

### Characterisation of convergence in Deligne-Mumford compactifiaction

1) Is there a (simple) way to characterise the convergence of complex curves toward a stable curve in term of hyperbolic metrics for the Deligne Mumford compactification of the moduli space of complex ...

**2**

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**0**answers

240 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 ...

**2**

votes

**0**answers

201 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 ...

**2**

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**0**answers

162 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 ...

**2**

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**0**answers

111 views

### Local curvature in a Cayley complex

I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local ...

**1**

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**0**answers

72 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 ...

**1**

vote

**0**answers

75 views

### Immersed surfaces in Hyperbolic 3-manifolds

Given a hyperbolic 3-Manifold $M=\Gamma_{0}\setminus\mathbb{H}^3$, and a smooth, connected, compact immersed negatively curved surface $\Sigma=\Gamma\setminus\widetilde\Sigma\subset M$, where ...

**1**

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**0**answers

49 views

### Complex structure and antipode map on the space of measured geodesic laminations

Fix a closed hyperbolic surface $S$, which represents a point in the Teichmüller space $\mathcal{T}$ of the underlying topological surface.
Thurston's earthquake theorem implies an identification ...

**1**

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**0**answers

81 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 ...

**1**

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**0**answers

80 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 ...

**1**

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**0**answers

112 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 ...

**1**

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**0**answers

67 views

### Coarse geometry of minimal surfaces in non-positively curved manifolds

Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface.
(You can basically image $X$ as a ball and $S$ as an embedded disk whose ...

**1**

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**0**answers

134 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 ...

**1**

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**0**answers

163 views

### Bounds on norm of harmonic function on degenerating hyperbolic surface

Suppose I have a Riemann surface with a shrinking geodesic which is degenerating towards a surface with a cusp, and I consider a neighborhood of the shrinking geodesic, $C_[a,b] = [a,b]\times ...

**1**

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167 views

### Measuring the complexity of a knot by minimum number of simplices to tile the complement

This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement
Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb ...

**1**

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**0**answers

230 views

### A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello,
This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :
If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...

**0**

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**0**answers

64 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 ...