**Edited 25.05.21: the assumptions of the question were incorrect**, but as the discussion may be helpful for future MOnauts, I'll strike my mistakes and add clearly marked explanations afterwards.

I am looking for a reason why a 3-manifold group $G$ that is virtually $\mathbb{Z}\times F$, $F$ being <s>either non-cyclic free or</s> a surface group, does not admit a presentation on two generators.

These are the fundamental groups of closed 3-manifolds with $\mathbb{H}^2\times\mathbb{R}$ geometry (**Added:** thanks @HJRW for pointing out that the strike-through case above corresponds to a non-empty boundary), and it turns out that all other geometries admit examples with fundamental group of rank two, with notable highlight of euclidean geometry where all fundamental groups are virtually $\mathbb{Z}^3$ (and rank two example being the Hantzche–Wendt/Fibonacci manifold). Thus the 3-manifold groups admit examples of virtually high rank groups being nonetheless of small rank themselves. Of course it is well known that a free group on two generators is virtually of arbitrarily high rank.

However, by [Boileau & Zieschang][1], Theorem 1.1, the rank of $\mathbb{H}^2\times\mathbb{R}$ manifolds depends on the genus of the base surface and number of singular fibers of the Seifert fibration <s>(and is at least 3)</s>, so apparently being virtually $\mathbb{Z}\times F$ forces the group to be of at least the same rank.

**Added:** this is my initial confusion - I assumed that the base orbifold of a $\mathbb{H}^2\times\mathbb{R}$ manifold must have genus at least 2, but this is not true. In fact, following the [Wikipedia's](https://en.wikipedia.org/wiki/Seifert_fiber_space) conventions for Seifert spaces, $\{-1,(o_1,0);(5,1),(5,2),(5,2)\}$ is a $\mathbb{H}^2\times\mathbb{R}$ manifold Seifert-fibering over a shpere, which in particular fits into Theorem 1.1, case ii) of the cited paper (just don't let the initial $g>0$ mislead you) and is indeed of both rank and genus 2. I thank again @HJRW for their comments which got me on the right track eventually. This of course makes the question that followed invalid.

<s>What is the cause that this subgroup bounds the rank of the ambient group from below and, say, free groups or abelian free $\mathbb{Z}^3$ do not? I would be happy if there is a geometric 3-dimensional reason in play here, but would be grateful for refreshing my general group theory as well.</s>


  [1]: https://link.springer.com/article/10.1007/BF01388469