Invariant Measures and Homeomorphisms of Boundaries

Measure classification is given for actions of the following type: 1. Any unbounded group Γ < G acting on G\H where G = PSL2(R) and H = Diff1(S1) 2. Any lattice Γ < G acting on G\H where G is as above and H = Homeoac(S1), the group of absolutely continuous homeomorphisms of the circle 3. Any uniform lattice Γ < G where G is a connected, center free, real Lie group of rank one, acting on G\H leaving some compact set Q invariant, where H = Homeo(G/P) where P < G is the unique (up to conjugation) proper parabolic subgroup of G. 4. Any real simple higher rankLie group G acting on H = Homeo(G/Q) where Q < G is any proper non-trivial parabolic subgroup. For each of these cases, it turns out that the only invariant probability measure on X = G\H is the Dirac measure on the identity coset.