Is it known if there are any examples of a finitely generated group $G$ such that:
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$G$ has a finite index subgroup $H$ which is free-by-cyclic
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$G$ itself is not free-by-cyclic
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$G$ is torsion-free
Since subgroups of free-by-cyclic groups are free-by-cyclic, one may strengthen (1) and ask that $H$ is normal in $G$. It is then fairly easy to construct groups that satisfy (1) and (2) by extending $H$ under any finite group. However, I couldn't come up yet with an example satisfying all three conditions. I've already know such a group must satisfy some properties:
- By a combination of Serre's and Stallings-Swan's theorems, such a group must have cohomological dimension 2.
- Since $H/[H,H]$ has a finite index image in $G/[G,G]$, $G$ must have infinite abelianization.
- In particular, $G$ admits homomorphisms onto $\mathbb{Z}$, all of whose kernels must have cohomological dimension exactly $2$. So $G$ must be a semidirect product $K \rtimes \mathbb{Z}$ for some group $K$ of cohomological dimension $2$.
Best Answer
The group $$G=\langle a, b, x, y\mid [a, b]^2=[x, y]^2\rangle$$ is a torsion-free group which is not free by cyclic. However, $G$ is free-by-$D_{\infty}$ and so virtually free-by-cyclic (containing an index-two subgroup which is free-by-cyclic).
This example is from the paper Baumslag, Fine, Miller and Troeger, Virtual properties of cyclically pinched one-relator groups. Int. J. Alg. Comp. (2009).