# Questions tagged [hyperbolic-geometry]

The hyperbolic-geometry tag has no usage guidance.

746
questions

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### Obstruction in construction of some lattices, related with $K3$ surfaces

I am considering a certain $K3$ surface that is lattice-polarized in two ways.
This leads to the following simple problem in lattice theory:
(Let me borrow notations for lattice from Ch.14 of this ...

4
votes

1
answer

99
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### Does every hyperbolic, almost-transitive, triangulation of $\mathbb{R}^n$ have boundary homeomorphic with $\mathbb{S}^{n-1}$?

Question 1: Let $T$ be a triangulation of $\mathbb{R}^n$. Suppose that the 1-skeleton of $T$ endowed with the graph-metric (i.e. each 1-cell is given length 1) is Gromov-hyperbolic. Suppose moreover ...

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### The figure eight knot complement in $S^3$

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...

2
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### Variants of Selberg trace formula

I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup ...

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### Inverse Laplace if products of hyperbolic function to other function [closed]

I want to calculate $\sinh(as) F(s)$. We have
$$L^{-1}\left [ \left ( \frac{e^{as}-e^{-as}}{2} \right )F(s) \right]=\frac{1}{2}L^{-1}\left ( e^{as} F(s)\right )-\frac{1}{2}L^{-1}\left ( e^{-as}F(s) \...

6
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1
answer

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### Are the non-free factors of Grushko decomposition of a finitely generated convex-cocompact (but not cocompact) subgroup of PSL$(2,\mathbb{R})$ finite?

See Grushko decomposition theorem.
Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?
...

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### Inverse limit in category of graphs

Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....

2
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1
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### Is every finitely generated classical Schottky group quasifuchsian?

$\DeclareMathOperator\PSL{PSL}$(Classical, finitely generated) Schottky groups are groups generated by finitely many hyperbolic elements of $A_i\in \PSL(2,\mathbb{C}), $ $i<n$ such that the ...

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3
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### Explicit triples of isomorphic Riemann surfaces

Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....

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### Convex core and geometric finiteness of negatively curved manifolds

I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...

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### Survey or good reference of taut foliations

I am interested in the topology of foliations.
In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows.
I guess that
A. Candel and L. Conlon, Foliations I (...

4
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1
answer

97
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### Complex length of geodesic added in hyperbolic Dehn surgery

Suppose $M$ is a cusped finite-volume hyperbolic $3$-manifold, say with a single cusp for simplicity. Following [NZ, Section 4] we can parametrize deformations of the hyperbolic structure with a ...

1
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1
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59
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### Are two nonsimple closed geodesics in minimal position?

We know that two simple closed geodesics are in minimal position, meaning that they realize the geometric intersection number. Is this result true for a pair of nonsimple closed geodesics?

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### Can every non-hemi uniform polytope tile hyperbolic space?

Can every uniform polytope which is not a hemipolytope tile hyperbolic space on its own?

4
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1
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### Does length spectrum determine a hyperbolic 3-manifold? What if we also know holonomies?

Question 1: Suppose that two hyperbolic 3-manifolds $M_1$ and $M_2$ with finitely generated fundamental group satisfy the property that for every closed geodesic in $M_1$, there is a closed geodesic ...

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### What are the volume-preserving diffeomorphisms of hyperbolic space?

What are the volume-preserving diffeomorphisms of $d$-dimensional hyperbolic space (in say the hyperboloid model)?
In particular, I'm especially interested in: what are the volume-preserving ...

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### Hyperbolic space embeds into Wasserstein space

Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...

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217
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### Induced homeomorphism from a quasi-isometry between hyperbolic spaces

Theorem. Let $\phi:X\rightarrow Y$ be a quasi-isometry between two (Gromov) hyperbolic spaces $X$ and $Y$. If $X$ and $Y$ are proper, then ϕ induces a homeomorphism between their boundaries.
The proof ...

3
votes

1
answer

161
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### Explicit formula for embedding Cayley graph of free group into hyperbolic space

The problem is to embed Cayley graph of free group with $n\geq2$ generators (the same as Bethe lattice with coordination number $2n$) into any model of $\mathbb{H}^2$ (we have no model preference, the ...

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1
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### Measured geodesic laminations have either discrete or Cantor set local cross-sections

I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076.
In section 1, after he defines measured geodesic laminations, he makes the ...

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2
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217
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### What is known about exceptional slopes of hyperbolic knots?

For $K$ a hyperbolic knot in $S^3$, a rational number $p/q$ is an exceptional slope if the manifold $M_{p/q}$ obtained from $(p,q)$-Dehn surgery on $K$ does not admit a hyperbolic structure.
Thurston ...

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### Computer verification for hyperbolic trigonometry

I am currently writing a paper that requires some lengthy computations using basic hyperbolic trigonometry. So, several hyperbolic figures appear, and we apply the law of sines and so on in order to ...

3
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1
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### Are geometric actions on CAT(0) spaces with isolated flats minimal on the boundary?

Suppose $X$ is a $CAT(0)$ space with isolated flats, $\partial X$ its visual boundary and $G$ acts properly discontinuously amd cocompactly on $X$. Must the $G$ action on $\partial X$ have a dense ...

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### Geodesic foliations of open manifolds foliated by hyperbolic spaces

It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete ...

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### For $f$ geodesically convex with $L$-Lipschitz-gradient on hyperbolic space, is $f(x) - f(x^*) \leq (const) \cdot L r$ for all $x \in B(x^*, r)$?

$\DeclareMathOperator\dist{dist}$Setting: Let $M$ be a hyperbolic space of sectional curvature $-1$, and let $f \colon M\to \mathbb{R}$ be a $C^2$, geodesically convex function which has $L$-Lipschitz-...

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### Doubly ruled surfaces in hyperbolic 3-space

A well-known theorem of classical surface theory states that the only doubly ruled surfaces in Euclidean 3-space are planes, 1-sheeted hyperboloids and hyperbolic paraboloids. There are a number of ...

2
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1
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### Name of the "s" parameter in Ungar's theory of hyperbolic geometry

I have done a R package which implements Ungar's approach to hyperbolic geometry, for the hyperboloid model. In this theory, there is a parameter $s>0$ which controls the curvature of the ...

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### Name of a foliation on an open subset of $\mathrm{PSL}(2,\mathbb R)$

To a representative $\left(\begin{array}{cc} a&b\\c&d\end{array}\right)$ of an element in $\mathrm{PSL}(2,\mathbb R)$ we associate
the point $\tau=\frac{b+id}{a+ic}$ of the Poincaré upper half-...

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1
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### Volume of hyperbolic 3-manifolds with toroidal boundary

A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.
This statement is from 3-Manifold Groups, page 18 (...

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### Monotonicity of perimeter of convex subsets of hyperbolic plane

I think that the perimeter of convex compact sets on the hyperbolic plane is monotone with respect to inclusion.
I am looking for a reference to the above fact.

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### Classification of principal monodromy elements

Let $(X,0)$ be a germ of normal analytic space with an isolated singularity at $0$, and let $Y:=X\backslash\{0\}$. Suppose $Y$ has a complex-hyperbolic metric which is complete at $0$. Burns-Mazzeo ...

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### Proof of homotopic essential simple close curves are isotopic

In the " A Primer on Mapping Class
Groups
Benson Farb and Dan Margalit"
We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\...

4
votes

1
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306
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### About isotopy and homotopy

In the " A Primer on Mapping Class
Groups
Benson Farb and Dan Margalit"
We have :
Proposition 1.10 Let $\alpha$ and $\beta$ be two essential simple closed curves in a surface $S$. Then $\...

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### Reference for shortest educational path to (Riemannian) hyperbolic plane

I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is consistent with Euclidean geometry. I would like to make an end-of-term ...

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### Number of small eigenvalues for flat unitary bundles

It is known that the number of small eigenvalues (eigenvalues less than $1/4$) of the Laplacian on hyperbolic surfaces may be topologically bounded from above. If $X$ is a finite area hyperbolic ...

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### Are hyperbolic $n$-manifolds recursively enumerable?

Fixing a dimension $n \ge 4$, is the class of closed hyperbolic $n$-manifolds recursively enumerable?
Since hyperbolic manifolds are triangulable I can reformulate this in the following more explicit ...

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### Fenchel-Nielsen length-length coordinates on Teichmueller space?

Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...

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### Example of maximal multicurve complex

in this paper we have :
" On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps."
Definition. The maximal multicurve complex $...

5
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1
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### Tzitzeica surface

A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface ...

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### Almost parallelizable hyperbolic manifolds

In Sullivan's paper Hyperbolic geometry and homeomorphisms (Proc. Georgia Topology Conf., Athens, Ga., 1977) he makes use of a closed hyperbolic almost parallelizable manifold in every dimension. ($M$ ...

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### Are there good mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry?

There are familiar analytic equiconsistency proofs for Euclidean and hyperbolic geometry. Those proofs are so robustly geometric that it seems like they must have synthetic analogues.
Looking into ...

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### Examples of the Thurston geometries with transitive Lie group action

Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries:
(1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup
(2) Euclidean: 3 torus $\...

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### Hyperbolic volume of hyperbolic knots

Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?
It seems that there is some necessary conditions:
$H_{1}(BG) = \mathbb{Z}$
$H_{2}(BG) ...

2
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### Counterexample to mostow rigidity theorem

I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not ...

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### Is there an effective genus theory for indefinite quadratic forms?

For positive definite quadratic forms, there is a way to check if two forms are isomorphic by arithmetic equivalence over $GL_n(Z)$ by computing configurations of vector up to some norm and then ...

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### Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...

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### Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach

I'm analyzing the following isometric immersion of $\mathbb H^2$ in $\mathbb R^\infty$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with
\begin{align}\label{5.1}
x_{2m-1}=\color{red}{2}\...

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### The best lower bound for isometric immersions

I just read Azov's article in the considered two classes of Riemannian metrics,
\begin{align*}
ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\
ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2
,&g>0\...

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### Variation of the geometry of a Dirichlet region as the defining point varies

Let $\Gamma$ a Fuchsian group acting on the hyperbolic plane $\mathfrak{H}$. For me, I am most interested in the case where $\Gamma$ has a fundamental domain that is a finite-polygon with all ...

0
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1
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### Half space vs growing balls in the hyperbolic plane [closed]

Let $p$ be a point in the $2$-dimensional hyperbolic space $H_2$. Consider a normal coordinate system $(x,y)$ at centered at $p$. Let $o_x\in H_2$ be the point of coordinate $(x,0)$ and let $C_x$ be ...