Which $F_3$-by-$\mathbb{Z}$s are CAT(0)?

Abstract

In this note we point out a mistake in theorem 4.4 of [Sam06], which states that a semidirect product $F_3\rtimes_\phi\mathbb{Z}$ whose defining automorphism $\phi$ is unipotent-polynomially-growing and fixes a free factor of rank $2$ is a CAT(0) group. We give and prove the corrected statement: such a group is CAT(0), if and only if $\phi$ is the identity or if the element of $F_2$ twisting the non-fixed generator is not in the commutator subgroup of $F_2$. This gives new examples of free-by-cyclic groups that cannot act properly by semisimple isometries on a CAT(0) space, that are similar to {Gersten}'s examples [Ger94]. We also construct CAT(0) structures for new examples of $F_3$-by-$\mathbb{Z}$s by thickening the strips in Bridson's tree of spaces construction [BH99].