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19.17 (2018)

Open

(Well-known problem). Suppose that $G$ is a finitely presented group such that the set of first Betti numbers over all the finite index subgroups of $G$ is unbounded above. (Here, the first Betti number is the maximum $n$ for which there is a surjective homomorphism onto $\mathbb{Z}^n$.)
$\qquad$ a) Must $G$ have a finite index subgroup with a surjective homomorphism to a non-abelian free group?
$\qquad$ b) Can $G$ even be soluble?

Contributor: J. O. Button

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