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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 …

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 …

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^ …

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 …

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 …

Let $X$ be a Riemann surface of class $P_G$ (i.e. which carries a Green function) but is Liouville (i.e. admits no nonconstant bounded harmonic functions). One way to construct these is to take a $\m …

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 …

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 …

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 …

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 $ …

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 …

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 …

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 $ …

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 …

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 …