Questions tagged [hyperbolic-geometry]
For questions about hyperbolic geometry, the branch of geometry dealing with non-Euclidean spaces with negative curvature, in which a plane contains multiple lines through a point that do not intersect a given line in the same plane.
912 questions
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Are the definitions of quasiconformal mappings using balls and spheres equivalent?
Let $(X,d)$ be a metric space and let $f:X\to Y$ be a homeomorphism between metric spaces.
In some sources the distortion of $f$ at a point $x\in X$ is defined using balls, e.g.
$$
H_f(x)=\limsup_{r\...
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On a closed hyperbolic surface, how do we know there exist infinite non-closed simple geodesics spiraling towards closed geodesics?
In Casson and Bleiler's Automorphisms of Surfaces after Nielsen and Thurston, they include a diagram
of an infinite non-closed simple geodesic on a closed hyperbolic surface, which limits on either ...
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Gluing boundaries of 3-manifolds to get hyperbolic 3-manifold
Suppose $M$ is a compact, connected, orientable, aspherical 3-manifold, whose boundary $\partial M=S_1\cup S_2$ is a disjoint union of two surfaces $S_1,S_2$ with the same genus $g>1$. Denote by $...
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Embedding of the Lobachevsky plane
Is this statement true? "The Lobachevsky plane whenever embedded in three dimensional Euclidean space takes the form a pseudosphere."
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Tiling the hyperbolic plane with non-regular polygons
Ref: Tiling the hyperbolic plane by non-regular quadrilaterals
Question: What is known about the tilings of the hyperbolic plane by n-gons that are not regular, especially for values of n greater than ...
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The intersection number of hyperbolic metrics as geodesic currents
Let $\Gamma$ be a Fuchsian group such that $\mathbb H^2/\Gamma$ is topologically a closed surface $S$. Bonahon notably introduced the space of geodesic currents $C(\Gamma)$ as the space of $\Gamma$-...
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Finite normal cover of closed hyperbolic manifold with bounded injectivity radius
Let $M$ be a closed (or finite volume) hyperbolic manifold that has injectivity radius $\leq l$. Does there exist a finite normal cover $p: \tilde{M} \to M$ such that $\tilde{M}$ has injectivity ...
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Exceptional isometries between modular curves
Given a subgroup $\Gamma\subset PSL(2,\mathbb{Z})$, let $H(\Gamma)$ be the set of $PSL(2,\mathbb{R})$-conjugates of $\Gamma$ which are contained in $PSL(2,\mathbb{Z})$, and let $h(\Gamma)$ be the ...
- 7,785
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Distribution of lengths of closed geodesics
(I am trying to get a sense of what the state of the art is regarding the distribution of the length spectrum of a closed surface of negative curvature, I am curious about any good reference/open ...
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Is the simplicial volume of a satellite realized by a representation?
For a link $L$ in $S^3$ let $S(L) = \| M_L, \partial M_L \|$ be the relative simplicial volume of the complement of $L$ times the volume of a regular ideal tetrahedron (here $M_L = S^3 \setminus \nu(L)...
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Does this algorithm generate all hyperbolic 3-manifolds with genus-2 boundary?
While studying the Euclidean path integral of 3D gravity with negative cosmological constant, my collaborators and I encountered a physically motivated algorithm that appears to generate a wide ...
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Asymptotic properties of Busemann functions on general metric spaces
I have been reading about Busemann functions on $\delta$-hyperbolic metric spaces from the book "Elements of Asymptotic Geometry" by Buyalo and Schroeder. Let $\partial_\infty X$ denote the ...
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Cusps of rank-one locally symmetric spaces
I've found in the literature these facts:
Any closed flat manifold is virtually (i.e. finitely covered by) a torus, and any finite-volume real hyperbolic manifold has virtually (i.e. is finitely ...
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Hyperbolic pants: boundary of collar neighborhood and shortest figure 8s
I'm looking for a construction of a $\mathbb{Z}/3$ symmetric pair of hyperbolic pants - all cuffs of length $a$ small, such that the shortest figure-8 geodesics in the surface have length bounded ...
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Is $\mathbb H^{p,q} _\mathbb C$ a symmetric space?
I asked a similar question yesterday and got negative votes. I was thinking I should ask the question in more details. I am very new to symmetric space stuff and if anyone knows this, please advise....
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