On Radon Measures Invariant Under Horospherical Flows on Geometrically Infinite Quotients.

We consider a locally finite (Radon) measure on |$ {\operatorname{SO}}^+(d,1)/ \Gamma $| invariant under a horospherical subgroup of |$ {\operatorname{SO}}^+(d,1) $| where |$ \Gamma $| is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not observe any additional invariance properties then it must be supported on a set of points with geometrically degenerate trajectories under the corresponding contracting |$ 1 $| -parameter diagonalizable flow (geodesic flow). [ABSTRACT FROM AUTHOR]