Cut numbers of 3-manifolds

The cut number of a manifold M, c(M), is the largest number of disjoint two-sided hypersurfaces in M which do not separate M. Equivalently, it is the largest rank of a free group being an epimorphic image of pi_1(M). We investigate the relations between the cut number and the first Betti number, b_1(M), of 3-manifolds M. We prove that the cut number of a ``generic'' 3-manifold is at most 2. This is a rather unexpected result since it is very hard to construct specific examples of 3-manifolds with with large b_1(M) and small c(M). On the other hand, we prove that for any complex semisimple Lie algebra g there exists a 3-manifold M with b_1(M)=dim g and c(M)<=rank g. Such manifolds can be explicitly constructed.