Horospherically invariant measures and finitely generated Kleinian groups (Q2665630)

Let \(\Gamma\) be a Zariski dense, discrete and finitely generated subgroup of \(G=\mathrm{PSL}_2(\mathbb C)\). The horospherical subgroup \(\mathrm{U}\) of \(\mathrm{PSL}_2(\mathbb C)\) is given by the upper-triangular unipotent matrices of \(\mathrm{SL}_2(\mathbb C)\). The paper is an attempt to understand the \(\mathrm{U}\)-ergodic and invariant Radon measures (e.i.r.m.) on \(G/\Gamma\). A measure \(\mu\) is called quasi-invariant with respect to an element \(g \in G\) if the action of \(g\) on \(G/\Gamma\) preserves the measure class of \(\mu.\) The author proves that any non-trivial \(\mathrm{U}\)-e.i.r.m. on \(G/\Gamma\) is \(N_G(\mathrm{U})\) quasi-invariant, where \(N_G(\mathrm{U})\) denotes the normalizer of \(\mathrm U\) in \(G\). The proof of this theorem uses a result by \textit{E. Lindenstrauss} and the author [Int. Math. Res. Not. 2022, No. 15, 11602--11641 (2022; Zbl 1503.37051)], together with the Tameness theorem by \textit{I. Agol} [Tameness of hyperbolic 3-manifolds, Preprint, \url{arXiv:math/0405568}] and \textit{D. Calegari} and \textit{D. Gabai} [J. Am. Math. Soc. 19, No. 2, 385--446 (2006; Zbl 1090.57010)].