I am a newcomer to hyperbolic geometry and was trying to understand some of it in the context of dynamics, for reading certain literature.
Let a discrete subgroup $G$ of $SL_2(\mathbb R)$ act on the upper-half plane $\mathbb H$ and consider the quotient $X$ as a hyperbolic space. Fix a point $x \in \mathbb H$. I would like a proof that the following are homeomorphic.
Gromov boundary of the group $G$;
The set of accumulation points of the set $\{gx: g \in G\}$ inside $\mathbb H \cup \mathbb R \cup \infty$.
Also: In each case, supposedly the result is indifferent of the point you start with. How to prove this?
Also, some standard books/articles for such topics will be appreciated. Easily accessible(preferably free) references would also be nice in addition.
Added: In the situation I was looking into, the group $G$ is either: free and convex co-compact, or: co-compact surface group. (Added for the case that this assumption makes things easier.)
In certain other situations, I was reading about similar dynamical results about boundary behavior in the case of more arithmetical groups but stated without notions such as Gromov boundary, and was led to wonder about the contrast. Therefore I am also curious about how different the behavior would be for more arithmetical groups; for example for $SL_2(\mathbb Z)$ and its congruence subgroups, and how would the Gromov boundary look like in those cases.
One more question: In the earlier version, I seem to have confused between the Gromov boundary of a group $G$ and the Gromov boundary of the space $X$. I hope my understanding of the latter is correct: The geodesic boundary in the sense of equivalence class of geodesics starting at $x$, and of bounded distance from each other as you go away from $x$. What exactly is the difference between the two? Some references that might help me in clearing this kind of basic misunderstandings also will be thankfully received.