I was studying Theorem 4.4.1 from John H. Hubbard's Teichmuller Theory, vol I, Theorem 4.4.1 ( P. 129 ) which states :


Let $X,Y$ be two hyperbolic Riemann surfaces with hyperbolic metrics $d\_X,d\_Y$ respectively and let $K\geq 1 $.Then there exists a function ( homeomorphism of positive real numbers) $\delta\_K:(0,\infty)\to(0,\infty)$ such that $\lim\_{\eta\to 0}\delta\_K(\eta)=0$ such that for all $K$-q.c maps $f:X\to Y$, we have $dist\_Y(f(x),f(y))\leq \delta\_K(dist\_X(x,y))$.


The way he proves it is the following : 1) It is enough to prove the statement for the universal cover ,i.e. the Poincare disk $D$, since a $K$-q.c. map lifts to a $K$-q.c map.


2) He defines $\delta\_K(\eta)= M^{-1}(\frac{1}{K}M(\eta) )$, where $M$ is the modulus of the branched/ramified cover with ramification locus being the two-point set $P={z\_1,z\_2}$, which ( modulus ) he proves depends only on the hyperbolic distance $dist\_D(z\_1,z\_2)$ . ( Lemma 4.4.2) and is a strictly decreasing homeomorphim of the positive real numbers.


My questions are :


1. Hubbard proves that the branched/ramified double cover of $D$ with ramification locus a two-point set is topologically a cylinder. But then how do we know that this cylinder has a finite modulus , i.e. the cylinder is not conformally equivalent to $C-{0},D-0 $ ? Well, in proposition 4.4.6 ( P. 132 ),he proves it, but that is only after proving Thm 4.4.1.
2. I am unable to follow the lines 4.4.2 and 4.4.3, in the proof of lemma 4.4.2 ? What does he mean exactly by his notation $ (D~\_r)\_ {0,z} $ ? 


Is he scaling the standard hyperbolic metric on $D$ ? And why exactly the inclusion of the cylinders ( ramified covers ) in the line 4.4.3 true ? Please explain , thanks !



Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )
1) As you said the first point is covered in prop. 4.4.6. anyway.


2) Concerning the second point, you can actually use that same proposition 4.4.6:


first the notation $d\_{D\_{r}}(0,z)$ means that you are considering the distance between $0$ and $z$ in the hyperbolic metric associated to the disk $D\_{r}$ centered at the origin with radius $r$ (so if you consider two such disks, $D\_{r\_{1}}$, $D\_{r\_{2}}$ you will work with two different metrics). 
Now you can normalize the situation so that your two points $z\_{1}, z\_{2}$ are $\pm a$ and use the double cover of $\mathbb{C}$ (ramified above $\pm a$) given by 
$$ \phi\_{a}(z)=(a/2)(z+1/z).$$ The preimage by $\phi\_{a}$ of a disk $D$ containing $\pm a$ will give you $\tilde{D}$ (because you obtain a double cover branched exactly where it should be). But since you have two such nested disks $D\_{r\_{2}} \subset D\_{r\_{1}}$ you will obtain an inclusion for their preimages as well $\tilde{D\_{2}} \subset \tilde{D\_{1}}$, as you wanted.


