Questions tagged [hyperbolic-geometry]
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Reference request: modulus of almost continuity?
Recently in some work I came across the following notion of a "not too discontinuous function". Let $(X,d)$ and $(Y,\rho)$ be metric spaces. For each $\delta\ge 0$ and every modulus of ...
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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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Critical exponent for groups with parabolics
I'm going to ask this question first in classical setting and then sketch its natural geometric setting.
Let $\Gamma$ be a subgroup of $\operatorname{PSL}_2(\mathbb Z)$ (the question is mostly ...
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Simple smooth functions on Bolza surface
Consider the Bolza surface, a compact Riemann surface of genus 2.
It is an octagon in the Poincaire disk model with opposite sides identified.
I would like to write down some analytic expressions for ...
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Figure 8 knot incomplete hyperbolic structure
The incomplete hyperbolic structure of the figure-8 knot $M$ is nicely reviewed in the notes by J.Purcell. The incomplete hyperbolic structure can be described by the holonomy representation of $\pi_1(...
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A sequence of conformal metrics with bounded negative curvatures on the disc
Let $\mathbb{D}$ denote the unit disk, and let $h_{-1}$ be the unique hyperbolic metric on $\mathbb{D}$ which is conformal to $dz^{2}$.
Take a sequence of smooth complete metrics $h_{n} = e^{\rho_{n}} ...
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On centrally symmetric convex figures on the hyperbolic plane
A planar region C such that there is an interior point that bisects all chords of C that passes through it may be termed centrally symmetric. It appears that such figures exist in non-Euclidean ...
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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)
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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 ...
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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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