Big deformations near infinity

In a related paper we showed that Ruelle's property for a Fuchsian group G fails if the group has a condition we called 'big deformations near infinity'. In this paper we give geometric conditions on R = double struck D sign/G which imply this condition. In particular, it holds whenever G is divergence type and R has injectivity radius bounded from below. We will also give examples of groups which do not have big deformations near infinity.