I was looking for the definition of the term "convex co-compact" in simple cases. But most references I find are looking into a little bit sophisticated notions such as mapping class group, Schottky subgroup and higher dimensional hyperbolic spaces. I would like to understand the definition just in the simple case of a discrete subgroup of $SL_2(\mathbb R)$ acting on the Poincaré half-plane.
What I could find was that the action is convex co-compact if the action is co-compact on the convex hull of the limit set $L$ . What I am doubting is the term "convex hull". Does this mean the collection of all geodesic segments connecting each pair of points in the set $L$ ?