I'm just get off the ground understanding some hyperbolic geometry / group theory and I came here to seek some help on some basic questions. In particular, I am reading through the hyperbolic estimates in the first sections of Cannon's "The Combinatorial Structure of Cocompact Discrete Hyperbolic Groups". Once the geometry comes into the picture, I seem to be doing alright, but I am having trouble understanding a couple compactness/point-set-topological arguments that come in at the beginning Cannon's article that I did not understand and I was hoping someone could point me in the right direction.
Here is the setting: $G$ is a finitely-generated subgroup of the isometry group of $\mathbb{H}^n$ for some $n$ with $\mathbb{H}^n/G$ compact and such that the action of $G$ on $\mathbb{H}^n$ is discrete (i.e., for every compact set $K \subset \mathbb{H}^n$, there are only finitely many $G$-translates of $K$ that intersect $K$). In addition, $C$ is a finite generating set for $G$ and $\Gamma$ is the Cayley graph of $G$ with respect to this generating set. Further, $\ast \in \mathbb{H}^n$ is a point on which $G$ acts freely.
(1) Why does such a point $\ast$ necessarily exist?
Using the action of $G$ on $\ast$ we then obtain a $G$-equivariant map $\phi : \Gamma \to \mathbb{H}^n$ that sends the edges to where they send $\ast$ and sends edges to hyperbolic geodesics. The following statements appear in the proof of Theorem 1.1 in the above paper and I do not understand / would like to understand the proofs:
(2) There exists a number $\alpha$ such that each point in $\mathbb{H}^n$ is within $\alpha$ of an element in $\phi(\Gamma)$.
(3) There exists a compact set $K \subset \mathbb{H}^n$ such that any two points in $\mathbb{H}^n$ of distance less than $2\alpha + 1$ lie in some $G$-translate of $K$.
Does anyone know how the proofs of these statements go? I do not at all know how to use the hypothesis that $\mathbb{H}^n/G$ is compact.