Direct product of free groups

Let $F_1,F_2$ be non-cyclic finitely generated groups. Then $F_1\times F_2$ is not amalgamated product or HNN extension over a cyclic subgroup.

Let $F_1,F_2$ be non-cyclic finitely generated groups. Suppose that $G=F_1\times F_2$ acts unboundedly, inversion-free, on a tree $T$ with cyclic edge stabilizers.

Suppose that each element of $F_1$ acts with a fixed vertex. Then $F_1$ has a fixed vertex (this is a general fact on tree actions of f.g. groups). Let $T_1$ be the set of vertices fixed by $F_1$. Then $T_1$ is a subtree, and is $F_2$-invariant. By minimality, we deduce that $T=T_1$, and hence $F_1$ acts trivially. Since $F_1$ is not cyclic, it follows that edge stabilizers are not cyclic, contradiction.

So some element $g_1$ of $F_1$ is loxodromic, with axis $A_1$. Similarly, some element $g_2$ of $F_2$ is loxodromic, with axis $A_2$. Since $F_2$ commutes with $g_1$, $F_2$ preserves $A_1$, and hence $A_2=A_1$, and since $F_1$ commutes with $g_2$, $F_1$ preserves $A_2=A_1$. So the action preserves an axis, hence by minimality $T$ is reduced to an axis.

Hence $G$ modulo a cyclic normal subgroup is trivial, of order $2$, infinite cyclic, or dihedral. This is excluded by the assumptions.