TITLE: When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?
QUESTION [22 upvotes]: The question has an easy answer, if one replaces free by free abelian: Then the resulting group is always solvable and a solvable subgroup of a CAT(0) group is virtually abelian.
If the resulting was CAT(0), then the chosen automorphism $\varphi$ in $\mathbb{Z}^n\rtimes_\varphi \mathbb{Z}$ would have finite order - otherwise the group would not be virtually abelian.
Now one can ask the same question for the free group instead or the free abelian group. 
I would like to know for which automorphisms $\varphi$ of the free group $F_n$ the group $F_n\rtimes_\varphi \mathbb{Z}$ is CAT(0).
I only know, that $F_n \times \mathbb{Z}$ is CAT(0). I think that if the chosen automorphism has finite order, then the result should be CAT(0) (although I don't have a proof). And I do not know automorphism, that gives a non-CAT(0) group.

REPLY [10 votes]: Seems worth recording here: there are some recent new examples of CAT(0) free-by-cyclic groups due to Rylee Lyman, see https://arxiv.org/abs/1909.03097v3. As she explains, "These are the first families comprising infinitely many examples for each rank of the nonabelian free group." Also, they're not relatively hyperbolic, which contrasts with the hyperbolic Hagen-Wise examples.