Ergodicity of Skew Product Action

Let (X, Ω, μ)be a nonatomic standard Borel space with probability measure μ. Let G be a locally compact second countable group with Haar measure λ. and with Borel field B. Let Γ be a countable automorphism group preserving the measure μ and acting ergodically in X by a Borel action. In Kim, it has been proved that supposing Γ is acting on (X, Ω, μ) ergodically, the skew product action (X ×a G, Γ) is ergodic if and only if the cocycle α has a dense range in G. Using this theorem, we have that the following theorem : Suppose that Γ is a Fuchsian group acting on a 2 - dimensioanl Riemannian manifold H of the constant sectional curvature - 1 and Hor(H) the space of horospheres in H. Then we have The Γ - action on the ideal boundary ∂H of H is ergodic and Γ-action on Hor(H) = ∂H × β R is also ergodic for some cocycle β : Γ × H → R.