# Is property FA of Serre known for $SL_2(\mathbb{Z}[i])$ and $SL_2(\mathbb{Z}[\zeta_3])$

In Serre's book Trees [Se, p. 68] it says:

3) For $$SL_2$$ the situation is different. It is clear that $$SL_2(\mathbf Z )$$ does not have property (FA). It is the same with $$SL_2(A)$$ when $$A$$ is the ring of integers of an imaginary quadratic field not isomorphic to $$\mathbf{Q}(\sqrt{- 1})$$ or $$\mathbf{Q}(\sqrt{-3})$$, since such a group has a quotient isomorphic to $$\mathbf{Z}$$ (cf. [20], th. 9, p. 519). On the other hand, if $$K$$ is an algebraic number field not isomorphic to $$\mathbf{Q}$$ nor to an imaginary quadratic field, one can show that every arithmetic subgroup of $$SL_2(K)$$ has property (FA); this applies notably to the groups $$SL_2(\mathbf Z[\sqrt{D}])$$ and their subgroups of finite index ($$D$$ a square-free integer $$> 1$$).

Question: Is it known whether $$SL_2$$ of the rings of integers of $$\mathbf{Q}(\sqrt{- 1})$$ and $$\mathbf{Q}(\sqrt{- 3})$$ has property FA?

[Se]: Serre, Jean-Pierre, Trees. Transl. from the French by John Stillwell, Berlin-Heidelberg-New York: Springer-Verlag. IX, 142 p. DM 48.00; {\$} 28.40 (1980). ZBL0548.20018.

Here is an answer for all Bianchi groups $$SL(2, O_d)$$: Such a group admits a nontrivial graph of groups decomposition (equivalently, does not have the property FA) unless $$d=3$$, in the latter case, it does not split, i.e. has the Property FA. For details, see: