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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 $L$ ?

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Yes, exactly. ${}$ – t.b. May 9 '11 at 13:36

Well slightly more than the union of all geodesics connecting points in L. You need to take the convex hull of this union. Think of a group G, a Scottish group generated by two hyperbolic isometrics. Start with four very small (euclidean) geodesics (1,2,3,4) in the Poincare unit disc model. Choose two hyperbolic isometrics: one identifying 1 with 2 and the other identifying 3 and 4. Remember that L will lie 'under' these small geodesics. If you only take the union of this example then you get 'holes', you get something that is clearly not a convex set!

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