An example of a free-by-cyclic group that is not CAT(0) was given by Gersten. It is constructed from the automorphism of $F_3\cong\langle a,b,c\rangle$ that sends

$a\mapsto a,~b\mapsto ba,~ c\mapsto ca^2~.$

The idea of the proof is to think about translation lengths and flats in any CAT(0) space on which it acts. As $\langle a,t\rangle\cong\mathbb{Z}^2$, it stabilises some flat. But $t$, $at$ and $a^2t$ are all conjugate, so have the same translation lengths. A little thought shows that this is impossible in a flat.

Note that, in many respects, (fg free)-by-cyclic groups are difficult to distinguish from CAT(0) groups. For instance, they have quadratic isoperimetric inequality.

An example of a free-by-cyclic group that is not CAT(0) was given by Gersten. It is constructed from the automorphism of $F_3\cong\langle a,b,c\rangle$ that sends

$a\mapsto a,~b\mapsto ba,~ c\mapsto ca^2~.$

The idea of the proof is to think about translation lengths and flats in any CAT(0) space on which it acts. As $\langle a,t\rangle\cong\mathbb{Z}^2$, it stabilises some flat. But $t$, $at$ and $a^2t$ are all conjugate, so have the same translation lengths. A little thought shows that this is impossible in a flat.

Note that, in many respects, (fg free)-by-cyclic groups are difficult to distinguish from CAT(0) groups. For instance, they have quadratic isoperimetric inequality.

As pointed out in this blog post, the question

Which free-by-cyclic groups are CAT(0)?

is Question 7.9 of Bridson's AIM article about `Problems concerning hyperbolic and CAT(0) groups'.