The stable type of the mapping class group and some relatively hyperbolic groups and applications to

The stable ratio set of a nonsingular action is a notion introduced by Bowen and Nevo to prove pointwise ergodic theorems for measure preserving actions of certain nonamenable groups.
I prove that stable ratio set of the action of a discrete subgroup of isometries of a CAT(-1) space with finite Bowen-Margulis measure on its boundary with the Patterson-Sullivan measure has numbers other than 0, 1 and \infinity, extending techniques of Bowen from the setting of hyperbolic groups.
I also prove the same result for the action of the mapping class group on the sphere of projective measured foliations with the Thurston measure, using some statistical hyperbolicity properties for the (non hyperbolic) Teichmueller metric on Teichmueller space.