In this paper we construct explicit examples of both closed and non-compact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the co-rank of a 3–manifold group (also known as the cut number) is bounded below by one-third the first Betti number.
"The co-rank conjecture for 3–manifold groups." Algebr. Geom. Topol. 2 (1) 37 - 50, 2002. https://doi.org/10.2140/agt.2002.2.37