On the cut number of a $3$–manifold

Abstract

The question was raised as to whether the cut number of a 3–manifold $X$ is bounded from below by $\frac{1}{3}{\beta }_1\left(X\right)$. We show that the answer to this question is “no.” For each $m\ge 1$, we construct explicit examples of closed 3–manifolds $X$ with ${\beta }_1\left(X\right)=m$ and cut number 1. That is, ${\pi }_1\left(X\right)$ cannot map onto any non-abelian free group. Moreover, we show that these examples can be assumed to be hyperbolic.