Commensurability and the character variety

Recall that hyperbolic 3-manifolds M and N are said to be commensurable if they have a common finite sheeted covering. This is equivalent to the fundamental groups having subgroups of finite index which are conjugate in PSL(2,C). In general it is very difficult to determine if two manifolds are commensurable or not, once the most obvious invariants of commensurability (for example, the invariant trace field, see [13] and [17]) agree. When M is a finite volume hyperbolic 3-manifold with a single cusp, its SL(2,C)-representation and character varieties, denoted respectively, by R(M) and X(M) throughout, have been fundamental tools in understanding the topology of M , see [6], [5], and [4]. These techniques can be extended to the PSL(2,C)-character variety of M , which we denote by Y (M) ([2], and see §2.1 for some details). Throughout, for either SL(2) or PSL(2), we use the subscript 0 to denote a component of X(M) (or Y (M)) containing the character of a faithful discrete representation of π1(M). The main results of this paper concern how Y0(M) be can used to detect incommensurability. For example, one of the main results can be summarized in the following (for terminology and definitions see §2):