On the Cut Number of a 3-manifold

الأصل · EN

The question was raised as to whether the cut number of a 3-manifold X is bounded from below by 1/3 beta₁(X). We show that the answer to this question is `no.' For each m>0, we construct explicit examples of closed 3-manifolds X with beta₁(X)=m and cut number 1. That is, pi₁(X) cannot map onto any non-abelian free group. Moreover, we show that these examples can be assumed to be hyperbolic.