# Questions tagged [hyperbolic-geometry]

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### Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?

It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure. Marc Peigné  constructed examples of geometrically infinite hyperbolic ...
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### Can a hyperbolic three-manifold have 𝑛 toric boundary components?

I'm wondering if I can construct a class of hyperbolic three-manifolds with $n$ toric boundaries. My idea is to take a bordered hyperbolic Riemann surface $\Sigma_{g,n}$, of genus $g$ with geodesic ...
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### Dirichlet domains of cusped hyperbolic surfaces

I'm used to think of cusped hyperbolic surfaces as equipped with ideal (all vertices are ideal) fundamental domain, but this is not in general a Dirichlet domain. Is is true that any such surface has ...
1answer
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### Is Tarskian hyperbolic geometry consistent, complete & decidable?

Tarski developed an axiomatic description of Euclidean geometry in first order logic. Its primitive notions are points and its primitive relations are betweeness and congruence of points. The Parallel ...
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### References on Hyperbolic Geometry and Teichmuller Theory

I am asking a soft question here. I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
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### Definition of quasi-geodesics

I was going through the books Géométrie et théorie des groupes by Michel Coornaert, Thomas Delzant, Athanase Papadopoulos and Metric Spaces of Non-Positive Curvature by Martin R. Bridson, André ...
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### References on Riemann surfaces

I have asked the question in MSE, but did not get an answer. I am asking a soft question here. I am interested in learning about Hyperbolic Geometry. I have read the book named "Fuchsian Groups&...
1answer
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### Dirichlet region of a free group

Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is ...
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### Geometrical meaning of a question from Marden

Let $T \in SL(2,\mathbb{C})$ be a normalised Möbius transformation. Then, $$|T(z) - T(w)| =|z-w||T'(z)^{\frac{1}{2}}||T'(w)^{\frac{1}{2}}|$$. The above is an exercise from Outer Circles by Marden (ex. ...
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### Maximum entropy distribution in the hyperbolic plane with given “mean” and “variance”

On the Euclidean sphere, by defining suitable analogs of the mean and variance in terms of the first circular moment, it can be shown that the von Mises distribution is the maximum entropy ...
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### Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?

For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
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### Covering hyperbolic manifolds by round balls

This question is a natural follow up to the question asked here. I think it should not be too hard to answer it (negatively) in dimension 3, but higher dimensions will be probably challenging. Let $M$...
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### Fixed point free involutions on Riemann surfaces

It is well known that a Riemann surface can have a fixed point free holomorphic involution only if it has odd genus. If it has one, is it unique? More generally, is any fixed point free automorphism ...
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### Exponential map and optimization

Apologies in advance for being somewhat vague. I'm trying to get pointers to establish a connection between a common trick used in practice in optimization, and the exponential map in differential ...
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### Absolute and relative tilings of the hyperbolic plane

In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it). The ...
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### What is sequential boundary of a $\delta$-hyperbolic space and how is the Gromov product extended to the boundary?

I have been reading up on $\delta$-hyperbolic spaces. But I am not getting a clear idea of sequential boundary of $\delta$-hyperbolic spaces and how the Gromov product is extended to it. Could ...
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### Complete geodesics on hyperbolic a pair of pants

I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here: I am trying to understand the article by Maryam ...
1answer
123 views

### Cusps of hyperbolic surfaces under finite covers

The following statement seems true, but I don't know a proof or a reference for it (and I would like one). Let $\Gamma< \operatorname{PSL}(2,\mathbb R)$ be a nonuniform lattice with one cusp. We ...
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### Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$Let $S$ be a compact oriented surface with nonempty boundary. There are two variants of Teichmuller space for $S$ you might consider: The ...
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### What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?

Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
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### An explicit (maybe algebraic) isometric embedding of the double torus with constant curvature K = -1

The following question is related to this previous question, Canonical immersion of the double torus: Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed ...
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### Number of Fuchsian groups with same trace field

Let $\Gamma,\Sigma\subset \mathrm{SL}_2({\mathbb R})$ be cocompact arithmetic subgroups. They are called commensurable in the wider sense, if there exists $g\in \mathrm{SL}_2({\mathbb R})$, such that ...