Trace fields of representations and virtually Haken 3-manifolds

Abstract

We introduce and study trace fields and invariant trace fields of SL2(C) and PSL2(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 1 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. In this paper a 3-manifold is always assumed to be connected and orientable, without loss of generality for the problems to be considered. A 3-manifold is said to be Haken if it is compact, irreducible, and contains a properly embedded incompressible surface. A 3-manifold is said to be virtually Haken if it has a finite cover which is Haken. The well-known virtually Haken conjecture states that every closed and irreducible 3-manifold with infinite fundamental group is virtually Haken. It is also conjectured that every closed Haken 3-manifold has virtually positive first Betti number, that is, the manifold has a finite cover which has positive first Betti number. If M is a closed hyperbolic Haken 3-manifold, then it is further conjectured that M has virtually 1 first Betti number, which means that for any positive integer n, there is a finite cover Mn of M such that the first Betti number of Mn is larger than n. These conjectures are fundamental and difficult issues in 3-manifold topology. In this paper we provide some information about these conjectures by studying traces of representations of 3-manifold groups into SL2ðCÞ or PSL2ðCÞ: For a group G and a representation r : G ! SL2ðCÞ; we define the trace field Kr of r to be the field generated by the traces of all the matrices r (g), g [ G; over the base field Q of rational numbers, that is, Kr ¼ Qðtrðr ðgÞÞ; g [ G Þ: