Schottky-type groups and minimal sets of horocycle and geodesic flows

In the first part of the paper the following conjecture stated by Dal'bo and Starkov is proved: the geodesic flow on a surface $M=\mathbb H^2/\Gamma$ of constant negative curvature has a non-compact non-trivial minimal set if and only if the Fuchsian group $\Gamma$ is infinitely generated or contains a parabolic element.
In the second part interesting examples of horocycle flows are constructed: 1) a flow whose restriction to the non-wandering set has no minimal subsets, and 2) a flow without minimal sets.
In addition, an example of an infinitely generated discrete subgroup of $\operatorname{SL}(2,\mathbb R)$ with all orbits discrete and dense in $\mathbb R^2$ is constructed.