On Radon measures invariant under horospherical flows on geometrically infinite quotients

We consider a locally finite (Radon) measure on $S{O}^{+}(d,1)\mathrm{/}\mathrm{\Gamma }$ invariant under a horospherical subgroup of $S{O}^{+}(d,1)$ where $\mathrm{\Gamma }$ 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).