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For general $p$, the only known method is to construct a Dirichlet fundamental domain and read off the group presentation from it. The procedure for computation of a fundamental domain is called "Jorg …

First, one should decide what quasiconvexity (qc) means in the context of subgroups $H\subset G$, where $G$ is merely a semihyperbolic group, e.g., a RAAG. (I am assuming that generating sets are fixe …

To make things interesting, I will consider two classes of manifolds (one strictly larger than the other) without compactness assumption: connected manifolds (dimension is finite but not fixed) admi …

Let $M$ be a compact hyperbolic 3-manifold and let $C(M)$ be the moduli space of flat conformal structures on $M$. The hyperbolic metric on $M$ defines a distinguished point $h\in C(M)$. One says tha …

These are difficult questions and very little in general is known about this. Mostly, what's known is ad hoc results for specific classes of manifolds (say, take some surgeries on 2-bridge knots...). …

There are various compact manifolds of negative curvature which are not homnotopy-equivalent to closed hyperbolic manifolds: Locally symmetric ones (complex hyperbolic, etc) as well as Gromov-Thurston …

The thing is that every compact Riemannian surface admits a $C^\infty$ isometric embedding in ${\mathbb E}^5$, see Michael Albanese's answer here; the result is a version of the Nash isometric embeddi …

This story is somewhat reminiscent of a (probably apocryphal) story of somebody writing a telegram to a mathematical research institute (Steklov Institute in Moscow, if I remember it correctly): "Sol …

B.Bowditch, Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113 (1993), no. 2, 245–317. The statement that you need is a corollary of his Proposition 5.6 in conjunction with finiteness …

Here is how you can define a hyperbolic plane over a finite field $F$, I do not know if it is sufficiently useful or interesting though. Let $V$ be a 3-dimensional vector space over $F$; let $Q$ be …

Here is the detailed answer. First, you have to assume that your hyperbolic manifold is complete and has finitely generated fundamental group, otherwise you will get no answer except for the tautologi …

First of all, I would use a different name for what you call geodesics on $X$, let's call them regular geodesics. The reason is that geodesics in Riemannian geometry are curves satisfying the equation …

There is such a proof given by Hjelmslev; it is based on a clever application of central symmetries (point-reflections). You can find it on pages 102-104 of Ф. Бахман, Построение геометрии на основе …

Let $G$ be a hyperbolic group with the Cayley graph $X$. Let $F<G$ be a finite subgroup and $L\subset \partial G$ be the fixed-point set of $F$. I will assume that $F$ is the maximal finite subgroup …

A good reference for this is, say, Kobayashi and Nomizu "Foundations of Differential Geometry". The result you are looking for is: If $M$ is a complete Riemannian manifold and $p: M\to M'$ is a (loca …