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:

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.


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