Fuchsian groups and compact hyperbolic surfaces (Q515310)

Let \(\Gamma_1\) be a finitely generated non-elementary discrete subgroup of \(G = \mathrm{PSL}_2(\mathbb{R})\) with no parabolics and let \(\Gamma_2\) be a cocompact lattice in \(G\). Then any \(\Gamma_1\)-orbit on \(\Gamma_2 \backslash G\) is either finite or dense. This is the first non-trivial case of a deep theorem of the first author and \textit{J.-F. Quint} [Ann. Math. (2) 174, No. 2, 1111--1162 (2011; Zbl 1241.22007)]. The authors' aim of the paper under review is to present a reasonably short and what they call rather elementary topological proof of this theorem. The proof is inspired by the recent work of \textit{C. T. McMullen} et al. [Invent. Math. 209, No. 2, 425--461 (2017; Zbl 1383.53044)] where they classify the orbit closures of the \(G\)-action on \(\Gamma_0 \backslash \mathrm{PSL}_2(\mathbb{C})\) for certain Kleinian subgroups \(\Gamma_0\) of finite covolume, which in turn is modeled after Margulis's original proof of the Oppenheim conjecture [\textit{G. A. Margulis}, Banach Cent. Publ. 23, 399--409 (1989; Zbl 0689.10026)]. The authors remark that their result also holds if there are parabolics, more precisely if \(\Gamma_1\) is any non-elementary discrete subgroup and \(\Gamma_2\) any lattice in \(G\). This follows from the paper of the first author and Quint [loc. cit.] cited above and the authors assert that the method of their paper can also be extended to this case.