Incommensurability criteria for Kleinian groups (Q2750912)

Two hyperbolic \(n\)-manifolds are called commensurable when they have isometric finite covering spaces. The main result of this paper is that any infinite collection of finite volume hyperbolic \(3\)-manifolds with uniformly bounded volume contains infinitely many commensurability classes. In fact, the collection can contain only finitely many manifolds from any commensurability class. The proof makes use of a result of Borel to show that the collection can contain only finitely many arithmetic manifolds, and a volume argument using a result of Margulis handles the nonarithmetic cases. Corollaries are that for each \(n\) the collection of all finite volume hyperbolic \(3\)-manifolds with exactly \(n\) cusps contains infinitely many commensurability classes, and the same for the collection of closed hyperbolic \(3\)-manifolds with fundamental group generated by two elements. The main result holds for \(2\)-manifolds as well, by the same argument. A version is also proven for hyperbolic \(3\)-manifolds of infinite volume; in this case the hypothesis is that the areas of the \(\Omega(G)/G\) are uniformly bounded, where \(\Omega(G)\) is the region of discontinuity of the fundamental group \(G\).