Are compact 3-manifold groups residually PSL(2,R) for finite, commutative R?

Montek Gill (Michigan)


It is known that the fundamental groups of compact 3-manifolds are residually finite. Can the class of finite groups in this statement be restricted to the groups PSL(2,R) for finite, commutative rings R? Feng Luo conjectured this to be the case after solving the hyperbolic gluing equations over commutative rings and thus constructing generalised holonomy representations. We provide a counterexample but also show that the conjecture does hold for many of the eight classes of geometric manifolds. This is joint work with Stephan Tillmann and Stefan Friedl.