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Here is the setup with which you are dealing: You have a smooth closed manifold $M$ and a finite cyclic branched covering $p: M\to M'$ ramified over a codimension 2 totally geodesic submanifold $V'\su …

Suppose that $G$ is a (not necessarily discrete) nonelementary group of isometries of the hyperbolic $n$-space. Then pairs of fixed points of loxodromic elements of $G$ are dense in $L(G)\times L(G)$, …

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 …

Let us say that an abstract group $\Gamma$ is self-contained (I just made up this terminology) if $\Gamma$ is isomorphic to its proper subgroup of finite index. Next, each discrete group of isometri …

I will start with few observations; along the way, I will correct your question. Let $g\in SO(n,1)$ be an orientation-preserving loxodromic element. Let $U_g\in SO(n)$ denote the rotational part of $g …

The number of cusps could be more than 1, see here, remark on page 294.

Here is a quick but nonelementary proof; however, if you are interested in geometry and topology of surfaces, or Teichmuller theory, you should learn about curve complex in any case, this is a very po …

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 …

I assume that $\varphi$ has finite order; a similar construction works in the infinite order case, just it is a bit harder. Consider $D$, the complement to the fixed point of $\varphi$. Then $S'=D/<\v …

Since the fiber is contractible, there is a global section. The details are in the papers Earle, Clifford J.; Eells, James, A fibre bundle description of Teichmüller theory. J. Differential Geometr …

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 …

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 …

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 …

For the second question the answer is obviously negative since distinct Euclidean lines cannot converge to each other (in the sense that the distance function goes to zero). For the first question, I …

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 …