Quasi-isometric rigidity for \(PSL_2 (\mathbb{Z} [1/p])\) (Q1976050)

It was proved by \textit{R. Schwartz} [Inst. Hautes Études Sci., Publ. Math. 82, 133-168 (1996; Zbl 0852.22010)] that every quasi-isometry of a nonuniform lattice \(\Lambda\) in a semisimple Lie group of rank \(1\) is equivalent to a unique commensurator of \(\Lambda \). The author of the paper under review calls this result commensurator rigidity. By\textit{B. Farb} and \textit{R. Schwartz} [J. Differ. Geom. 44, 435-478 (1996; Zbl 0871.11035)] it was conjectured that commensurator rigidity, or at least a slightly weaker statement, ``quasi-isometric if and only if commensurable'', should apply to nonuniform lattices in a wide class of Lie groups. In the paper under review it is proved that both these statements are true for \(PSL_2(\mathbf Z[1/p])\).