# Two questions from Hubbard's Teichmuller theory book Vol I, P. 130 , Thm 4.4.1, ( QC maps )

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 !

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.