The hyperbolic-geometry tag has no usage guidance.

**16**

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438 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, ...

**12**

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**0**answers

204 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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228 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**

votes

**0**answers

528 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

241 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
...

**9**

votes

**0**answers

384 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**

votes

**0**answers

115 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 K_3^{\operatorname{...

**8**

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**0**answers

205 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**

votes

**0**answers

162 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 ...

**7**

votes

**0**answers

80 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 ...

**7**

votes

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112 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 ...

**6**

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**0**answers

144 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**

votes

**0**answers

181 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**

votes

**0**answers

327 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

621 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

239 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

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 ...

**5**

votes

**0**answers

87 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**

votes

**0**answers

273 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

107 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**

votes

**0**answers

151 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 ...

**4**

votes

**0**answers

142 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 ...

**4**

votes

**0**answers

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) ...

**4**

votes

**0**answers

68 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**

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**0**answers

166 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 $3$--...

**3**

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**0**answers

120 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

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) ...

**3**

votes

**0**answers

82 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**

votes

**0**answers

141 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

83 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 importance,...

**3**

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**0**answers

156 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

318 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**

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**0**answers

161 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

48 views

### automorphic forms associated with symmetries of vertices of uniform honeycombs in hyperbolic space

Is there a catalogue of automorphic forms (modular/Maass/Siegel/Hilbert...) which lists them in terms of Poincaré series associated with the symmetries of the vertices of uniform honeycombs in ...

**2**

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**0**answers

52 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} &...

**2**

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**0**answers

48 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 ...

**2**

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**0**answers

175 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**

votes

**0**answers

87 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 x,y,z\...

**2**

votes

**0**answers

74 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

72 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

250 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

212 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

164 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

108 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**

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**0**answers

49 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 ...

**1**

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103 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 X\to\...

**1**

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**0**answers

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 ...

**1**

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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 ...