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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Siegel domain associated to a symmetric space
Consider the symmetric space
$$
\frac{\mathrm{SU}(2,1)}{\mathrm{S(U}(2) \times \mathrm{U}(1))}= \mathbb{H}_\mathbb C^2.
$$
For this symmetric space there is a Siegel domain (See the page 22 of Complex ...
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Exponential divergence of geodesics in Gromov hyperbolic spaces
Let $X$ be a Gromov-hyperbolic geodesic metric space. I am willing to add additional assumptions (properness?) but for now I will be as general as possible. I would like to know the (strong) ways one ...
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Flat bundles on hyperbolic manifolds
Let $G$ be a discrete and torsion-free subgroup of $\mathrm{O}^+(n,1)$ and let $\rho\colon G \to \mathrm O(k)$ be an orthogonal representation. As I understand, we can construct a flat vector bundle $\...
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Hyperbolicity of certain 3-manifolds with genus-two boundary
Consider the table of handlebody-knots given by Ishii, Kishimoto, Moriuchi, and Suzuki. By embedding each of these knots in $S^3$ and removing a tubular neighborhood of the knot, we obtain a 3-...
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Recursive formula on dimensions for integrating the Gaussian on d-dimensional hyperboloid
Are there any dimension reduction formulas or recursive formula on dimensions for integrating the Gaussian function on d dimensional hyperboloid by using an integration formula on a d-2 dimensional ...
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Are the free and surface group von Neumann algebras isomorphic?
This post builds upon the previous question Is there a pair of non-isomorphic torsion-free hyperbolic groups that have isomorphic von Neumann algebras?, by exploring a specific case, as suggested by ...
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Simple closed geodesics on modular curves
Let $H$ be the upper half plane. For a finite index subgroup $\Gamma\le PSL(2,\mathbb{Z})$, the closed geodesics on the modular curve $H/\Gamma$ correspond to conjugacy classes of cyclic subgroups ...
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Is there a pair of non-isomorphic torsion-free hyperbolic groups that have isomorphic von Neumann algebras?
By the uniqueness of the hyperfinite ${\rm II}_1$ factor [Co76], the group von Neumann algebra $ L(G) $ is isomorphic to $ L(S_{\infty}) $ for every ICC amenable group $ G $.
In contrast, Popa's ...
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Basic questions about visual hyperbolic metric spaces
I have a couple of basic questions about visual hyperbolic metric spaces. A visual hyperbolic metric space $X$ is a hyperbolic metric space with the property that there exists an $o \in X$ such that ...
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A curious square relating models of hyperbolic space
One of my favorite tricks in hyperbolic geometry is the inclusion of the $3$-dimensional hyperboloid $\mathbb{H}^3:=\{-x_0^2+x_1^2+x_2^2+x_3^2=-1, x_0>0\}$ into the space $\operatorname{Herm}(2)$ ...
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Geometric Optics in Hyperbolic geometry
Assuming Fermat's principle that light always chooses least time trajectories, what are the trajectories in Hyperbolic geometry? For example, is there any change in Snell's law of refraction in ...
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The uniqueness of constant scalar curvature metrics in the conformal class of a hyperbolic metric
In the sphere, the constant scalar curvature metric may not be unique within the conformal class of a metric. What happens in a closed hyperbolic $n$-manifold ($n>2$)? Can the fact that the Yamabe ...
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Examples of knots with a high-dimensional component of the character variety
For a knot $K$ let $\mathcal{X}(K)$ be the $\operatorname{SL}_2(\mathbb{C})$ character variety.
We say it is high dimensional if there's a component with dimension $> 1$; one easy way to find such $...
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Metric on an embedded product of two hyperbolic planes
The product of two copies of $H^2:=\{z\in\mathbb{C}\mid \operatorname{Im}(z)>0\}$ can be embedded into $\mathbb{P}^3$ via the Segre embedding
$$
\begin{array}{ccc}
\sigma: & \mathbb{P}^1\times\...
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Is rank of the length spectrum of a closed negatively curved surface/manifold infinite?
Suppose that $(S,\mathfrak{g})$ is a closed negatively curved Riemannian surface (or more generally a manifold). Negative curvature guarantees that the non-trivial conjugacy classes $\text{conj}(\pi_1(...
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Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
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Connectedness of the space of negatively curved metrics of a compact 3-manifold
Is the space of metrics of negative sectional curvature over a closed 3-manifold connected? If so, in what paper is this result stated?
Note: as the Ricci flow hyperbolizes negatively curved metrics, ...
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Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
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Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
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Unitary representations of Fuchsian and Kleinian groups
Let $\Gamma$ be a discrete group that is either Fuchsian ($\Gamma \subseteq \text{PSL}(2,\mathbb R)$) or Kleinian ($\Gamma \subseteq \text{PSL}(2,\mathbb{C})$).
I have a unitary representationL
$$
\...
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When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
user519738
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The length is bounded
Let $\Sigma$ be a surface of finite type. Let $\mathcal{S}$ be the set of non-trivial isotopy classes of simple closed curves on $\Sigma$. One denotes by $l_x(\alpha)$ the infimal length of curves in ...
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Existence of orientable finite volume complete cusp hyperbolic 3-manifolds $\mathbb{H}^3 / \Gamma$, where $\Gamma$ has no parabolic generators?
Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret ...
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Existence of a geodesic on a non-orientable surface
Let $\Sigma$ be a non-orientable surface possibly with boundary or punctures. Is it possible that a one-sided loop in $\Sigma$ is always realized as a geodesic?
In the orientable case, it is well-...
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Geodesic convexity of Dirichlet Fundamental Domains
My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer:
Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a ...
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Is the circumcenter Lipschitz on large convex sets in hyperbolic space?
Given a uniquely geodesic metric space $X$, let $\mathcal K(X)$ denote the metric space of compact, convex subsets of $X$ equipped with the Hausdorff distance. Given $K \in \mathcal K(X)$, let $c(K)$ ...
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Projections of closed geodesics under the modular function
In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...
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Space of thick ending laminations
Let $\Sigma$ be a compact closed connected oriented surface of genus $g>1$. Klarreich proved that the space of ending laminations $\mathcal{EL}(\Sigma)$ is the ideal boundary of the curve complex $...
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Can I endow the following 3-manifold with a hyperbolic metric?
Consider the following three-dimensional topology. Start with $S^3$ and drill out four unlinked tori as shown in the picture. Then, fill in the gaps with the same tori but with their longitudes and ...
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Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space
Consider the group $\operatorname{PSL}(2,\mathbb C)$ acting by Möbius transformations of the Riemann sphere. It is known that this action can be extended to an action on the unit ball which is ...
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Understanding a "straightforward" application of the method of stationary phase for proving a trace formula on compact hyperbolic surfaces
I'm reading the paper 'Uniform distribution of eigenfunctions on compact hyperbolic surfaces' by Steven Zelditch (Duke Mathematical Journal, 55, pp. 919-941 (1987), MR916129, Zbl 0643.58029) and am ...
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Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
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If $X$ is a hyperbolic, locally finite graph with $\partial X \cong S^1$, and $G$ acts cocompactly but not properly on $X$, what can we say?
It is an important and deep fact of geometric group theory that if the Gromov boundary of a hyperbolic group $G$ is a circle, then $G$ is virtually Fuchsian [Tukia, Gabai, Casson-Jungreis...]. I am ...
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When do two measured foliations on a surface define a Riemann surface structure?
Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...
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MGFs of sum of (Rademacher) independent variables and (hyperbolic/spherical) Pythagorean theorem
Consider a set of iid random variables $X_1, X_2, \ldots$ (distribution to-be-specified later). For real numbers $a_1, a_2, \ldots$ (with $\sum_{k} a_k^2 < \infty$) define
$$X = a_1 X_1 + a_2 X_2 +...
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