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Indeed, the hyperbolic plane is $\log(2)$-hyperbolic (with the 4-point definition of hyperbolicity) and this is the optimal constant. The result is nontrivial and first appeared as Corollary 5.4 in Ni …

As for conformal moduli of the tori (more precisely, Teichmuller parameters) that appear: It is hard to tell, afaik, there is no explicit description. We know that these will be elements of $\bar{{\ma …

Here is an answer of sorts; it is not completely canonical though. First of all, you have to pick a conformal or hyperbolic structure on the fiber $\Sigma$. This can be made almost canonical, since th …

Your conjecture is false. Every nonorientable closed connected surface of negative Euler characteristic, admits a hyperbolic metric such that the surface is covered by 3 embedded disks. Hence, for eac …

Let $S$ be a complete hyperbolic surface of finite area, $f: S\to S$ be a pseudo-Anosov homeomorphism which lies in a torsion-free finite index subgroup $\Gamma$ of the mapping class group $Mod_S$ (fo …

Just to remove this question from the un-answered list. First of all, if $d_{GH}(X,Y)\le \epsilon$ then there is a $(1,2\epsilon)$-quasi-isometry $X\to Y$. See for instance Burago, D.; Burago, Yu.; Iv …

Try sections 1-15 of this paper: Cannon, James W.; Floyd, William J.; Kenyon, Richard; Parry, Walter R., Hyperbolic geometry, Levy, Silvio (ed.), Flavors of geometry. Cambridge: Cambridge University P …

For the $i$th factor ${\mathbb H}^2$ in the product of hyperbolic planes, pick a complete geodesic $c_i$, $i=1,...,k$. The product $$ F=c_1\times ... \times c_k\subset X=\prod_{i=1}^k {\mathbb H}^2 $ …

The question is what to modify and how much one is willing to modify. As everybody who seriously thought about elementary geometry knows, Euclid's axioms are inadequate. There are several "standard" a …

Here is what I think is the correct setup: Let $X$ be a symmetric space of noncompact type, $\Gamma$ is a lattice in the isometry group of $X$. Then $\Gamma$ has finitely many $\Gamma$-conjugacy class …

In order to remove this question from the "unanswered list." Let $\epsilon>0$ be the Margulis constant for the hyperbolic plane (with curvature $-1$). Then for every complete hyperbolic surface $S$, i …

This is an answer to questions 7 and 8 (I have to say, having 8 questions in one post is way too much for my taste): Suppose that $M$ is a finite-volume quotient of $H^3$ or a compact quotient of $H^ …

I do not have a self-contained reference, but the key is Long, D. D.; Reid, A. W., Constructing hyperbolic manifolds which bound geometrically, Math. Res. Lett. 8, No. 4, 443-455 (2001). ZBL0992.57023 …

I am not sure how well you know the Teichmuller theory, but the basic thing to understand is that $\mathcal T$ is not the space of hyperbolic metrics on $\Sigma$: It is the space of pairs $(\sigma, [\ …

Let $\Gamma< PSL(2,\mathbb R)$ (not $SL(2,\mathbb R$)!) be a cocompact Fuchsian group. Given a Lie group $G$ one defines The representation variety $R(\Gamma, G)=Hom(\Gamma,G)$. One further defines $ …