 The statement on Lebesgue measures follows for classical Schottky groups from the existence of the Poincaré series

Poincaré showed that the series | "c""i" |−4 is summable over the non-identity elements of the group.In fact taking a closed disk in the interior of the fundamental domain, its images under different group elements are disjoint and contained in a fixed disk about 0.So the sums of the areas is finite.