Horospherically invariant measures on geometrically infinite quotients.

We consider a locally finite (Radon) measure on $\mathrm{SO}^+(d,1) / \Gamma$ invariant under a horospherical subgroup of  $\mathrm{SO}^+(d,1)$ , where $\Gamma$ is a discrete but not necessarily geometrically finite subgroup. We show that whenever the measure does not admit 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). We deduce measure classification results. One such result in the context of finitely generated Kleinian groups in $\mathrm{PSL}_2(\mathbb{C})$ will be highlighted. Most of the talk will be based on joint work with Elon Lindenstrauss.