On Radon Measures Invariant Under Horospherical Flows on Geometrically Infinite Quotients

We consider a locally finite (Radon) measure on SO+ (d, 1)/ invariant under a horospherical subgroup of SO+ (d, 1) where is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not observe any additional invariance properties then it must be supported on a set of points with geometrically degenerate trajectories under the corresponding contracting 1-parameter diagonalizable f low (geodesic flow).