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### 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 ...

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398 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, ...

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217 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
...

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392 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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340 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.
...

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141 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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188 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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58 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 ...

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108 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 ...

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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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292 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 ...

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341 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 ...

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194 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 ...

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**0**answers

143 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 ...

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**0**answers

112 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 ...

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**0**answers

67 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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125 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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141 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 ...

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**0**answers

258 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 ...

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151 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 ...

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**0**answers

46 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 ...

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187 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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**0**answers

134 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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133 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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109 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 ...

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53 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 ...

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125 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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140 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 ...

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153 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 ...

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205 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 ...

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24 views

### 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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71 views

### 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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56 views

### Geometric effects of removing elements of D2n generalizable?

So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a ...

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145 views

### Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...

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195 views

### Mapping from $\text{PSL}(2,\mathbb{R})$ to transformations of the hyperboloid

Let $F$ map points in $\mathbb{R}^3$ to points on the unit disk, $\Delta$, in the $xz$-plane (identified with $\mathbb{C}$) by projecting through $\Delta$ along lines that intersect at $(0,-1,0)$. ...

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159 views

### Quick references/sources for the hyperbolic Riemann Surfaces with boundary

Hello,
Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of ...

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98 views

### uniform properness of lifts of uniform proper maps

Let $(X,d)$ be a unique geodesic metric space and $Y\subseteq X$ a subset, equipped with the induced path metric $d'$. We say that the inclusion $i\colon Y\hookrightarrow X$ is uniformly proper if ...

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246 views

### How many pants decompositions for a given hyperbolic surface with a given hyperbolic metric

Given a hyperbolic surface of genus g ( >= 2 )and given a fixed metric on it, how many pants decompositions exist for that surface? I tend to believe that it is finite ? For example, if we take a ...