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](https://zbmath.org/?q=an:0548.20018).



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


C. Frohman and B. Fine, Some Amalgam Structures for Bianchi Groups, Proceedings of the American Mathematical Society, Vol. 102, No. 2 (1988), pp. 221-229. 


