SAT Actions and Ergodic Properties of the Horosphere Foliation

The notion of a SAT action introduced by Jaworski was inspired by the proximal properties of the Poisson boundaries of random walks obtained by Furstenberg and by the definition of an approximatively transitive action due to Connes and Woods. We introduce a slightly stronger condition SAT* and establish several general ergodic properties of SAT* actions. This notion turns out to be quite helpful for studying ergodic properties of the horosphere foliation (particular case: the horocycle flow) on a quotient of a CAT(-1) space by a discrete group of isometries G. We look at the “intermediate covers” determined by normal subgroups H ⊲ G. Under the condition that the boundary action of G is SAT* we establish conservativity of the horosphere foliation on these intermediate covers and prove that its ergodic components are in one-to-one correspondence with the ergodic components of the boundary action of H. In particular, in this situation ergodicity of the horosphere foliation is equivalent to ergodicity of the associated boundary action. These results are applicable to several well known classes of measures at infinity (harmonic, Gibbs, conformai).