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12.41 (1992)

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Let $F$ be a free group on two generators $x, y$ and let $\varphi$ be the automorphism of $F$ defined by $x \mapsto y$, $y \mapsto xy$. Let $G$ be a semidirect product of $F/(F''(F')^2)$ and $\langle \varphi \rangle$. Then $G$ is just-non-polycyclic. What is the cohomological dimension of $G$ over $\mathbb{Q}$? (It is either 3 or 4.)

Contributor: P. H. Kropholler

Progress

It is 4, because $G$ is not constructible, and then its cohomological dimension is equal to the homological dimension plus 1 by Theorem III.8 in (M. R. Bridson, P. H. Kropholler, J. Reine Angew. Math., 699 (2015), 217–243), while the homological dimension is equal to the Hirsch length by (U. Stammbach, J. London Math. Soc. (2), 2 (1970), 567–570).

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