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We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$ -manifold. Our estimates are polynomials in the tight areas and Bowen–Margulis–Sullivan densities of geodesic planes, with degree given by the modified critical exponents. 

Abstract Let $\Gamma $ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to ${\mathbb {R}}^{\text {rank}\,G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup that normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures. 

Abstract Let M be a geometrically finite acylindrical hyperbolic $3$ -manifold and let $M^*$ denote the interior of the convex core of M . We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J. , to appear, Preprint , 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $3$ -manifold $M_0$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $M_0$ . We construct a counterexample of this phenomenon when $M_0$ is non-arithmetic.