# Questions tagged [hyperbolic-geometry]

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### The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions

I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...
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### Thurston's 24 questions: All settled?

Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":       $\cdots$ Two naive questions from an outsider: (1) Have all $24$ now been resolved? (2)...
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### can you fool SnapPea?

A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with. What I'm looking for is a non-hyperbolizable knot ...
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### Geodesics on the twisted pseudosphere (Dini's surface)

I wonder how difficult it is to compute geodesics on Dini's Surface, a twisted pseudosphere? Here is one parametrization, from Alfred Gray's Modern Differential Geometry of Curves and Surfaces, p....
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### Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?

In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...
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### Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
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### Non-residually finite matrix groups

By Malcev's theorem, every finitely generated linear group is residually finite (RF). On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to ...
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### Reference request: Geodesic flow on a manifold with negative curvature is ergodic

I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result: The geodesic flow on a manifold with negative curvature is ergodic. The lecture note that ...
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### Maximum of a function of one variable

Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...
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### Canonical immersion of the double torus

It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
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### Immersions of the hyperbolic plane

Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples? Edit: Although I did not originally say so, I was ...
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### Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact: Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is spherical/euclidean/...
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### Smallest tile to tessellate the hyperbolic plane

Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself. I think it will be a Triangle group, but I'...
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### Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?

Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing ...
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### Bolyai's construction

Here is Bolyai's construction (in Klein model), which I learned recently from answer of Will Jagy to this question. It is a Compass-and-straightedge construction of asymptotically parallel line in ...
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I try to understand some of the topology of the space of pointed non-compact hyperbolic surfaces (with the pointed Gromov-Hausdorff topology). It is known that the fundamental group of a non-compact ...
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### Canonical fundamental domain for a discrete subgroup Γ of SL₂(R) acting on hyperbolic plane

Let a discrete subgroup $\Gamma$ of $SL_2(\mathbb R)$ act on the hyperbolic plane by Möbius transformations. Is there a "best" or "most canonical" fundamental domain for this action? Some (mostly ...
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### Geodesics on a hyperbolic paraboloid

Given any two points on a hyperbolic paraboloid ($xy = z$ or $z = (x^2 - y^2)/2$) how does one find the geodesic between them? I know that since the hyperbolic paraboloid is doubly ruled, some of the ...
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### Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry

Hello, Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also ...