# Questions tagged [hyperbolic-geometry]

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### Number of regions created by r hyper-planes in n-dimensional space [closed]

I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r) ...
1 vote
102 views

### Re: Vertices of hyperbolic triangle with given angles [closed]

I'm working on visualizing hyperbolic triangles given angles following a previous discussion on MathOverflow. The algorithm, as outlined in this MathOverflow answer, involves computing the side ...
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### Residual finiteness of hyperbolic 3-manifold groups

So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is: Q1. If $M$ is an infinite-...
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### Hyperbolic three-manifolds that fiber over the circle

Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...
1 vote
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### Describing the hyperbolic structure of punctured torus in terms of the period lattice

Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$. ...
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1 vote
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### Divergence functions in hyperbolic groups

Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below. We note that in $\mathbb{R}^2$ there is no divergence ...
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### Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$ I have a reference request for a proof for the following statement in the title: The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
110 views

### Theorems that hold in Euclidean and Elliptic Geometry but not in Hyperbolic (and vice versa)

Question: Which theorems (if any) of plane Euclidean geometry continue to hold in Elliptic geometry but don't hold in Hyperbolic? And are there theorems that are valid in Euclidean and Hyperbolic ...
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### Solving equations in hyperbolic groups and subgroups of isometry of a Gromov hyperbolic space

Let $\Gamma$ be a hyperbolic group. Let $g$, $\gamma\in \Gamma$ freely generate a non-abelian semigroup (in particular, they don't commute and have infinite order). Does the equation $g\gamma^n=h^m$ ...
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### Is every uniform hyperbolic linear space infinite?

I start with definitions. Definition 1. A linear space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ satisfying three axioms: (L1) for any distinct ...
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### Relation of geometric and polyhedral convergence

By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...
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### Minimum diameter of spherically-inverted topological balls

Let $U$ be the closed unit ball in $\mathbb{R}^3$. Let $S$ be a round sphere whose center is in $U$ with radius at least $\delta_1 > 0$. Suppose $B$ is a closed topological ball of Euclidean ...
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### Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul

In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
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### Markov property for groups?

My question again refers to the following article: Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...
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### Why can't a Lie group act transitively on a finite volume hyperbolic manifold?

In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?", it is claimed that if $M$ is a manifold which admits a finite volume hyperbolic metric (...
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### Geodesics in free homotopy classes and the fundamental group

Let $\mathcal{H}$ be the upper half-plane and $\Gamma$ be a cocompact, torsion-free Fuchsian group. The quotient space $X=\Gamma\backslash \mathcal{H}$ is a smooth closed Riemann surface and there is ...
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### Is the canonical map from isometry group of a Gromov hyperbolic space to homeomorphisms of its Gromov boundary injective?

Suppose X is a proper Gromov hyperbolic space and $\partial X$ is its Gromov boundary. It is well-known that there is a canonical group homomorphism $\Phi$ from the isometry group of X to the group ...
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