Trace fields of representations and virtually Haken 3-manifolds

We introduce and study trace fields and invariant trace fields of SL"2(C) and PSL"2(C) representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or ∞ first Betti number. In particular, we define the notion of an algebraically trace-proper surface subgroup in a 3-manifold group, and show that any closed orientable irreducible 3-manifold with such a surface subgroup is virtually Haken. We give infinitely many families of closed orientable hyperbolic non-Haken 3-manifolds with algebraically trace-proper surface subgroups.