On Radon Measures Invariant Under Horospherical Flows on Geometrically Infinite Quotients

Abstract

Abstract 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).