While not so well known, the von Neumann paradox is built among the same lines, in dimension 2 and with transforms within the special linear group. But what is wrong with the following "proof" : it is not very hard to show that the group generated by the two rotations of angle $\alpha$ around $O$ and $C$, with $OC<1/100$, say, and $\alpha/\pi$ irrational and small (say $<1/100$ too), is nonabelian and free. The von Neumann construction applied to two disjoint unit discs then split them in four sets (plus a few fixed points) and sends those sets injectively, by rotations and translations, to disjoint sets included in the union of four copies of the unit disc, this union having something like 1.04 area of the disc. Obviously, this is very wrong, as the BanachTarski construction of a full additive measure invariant by isometries of the plane preclude such a thing. Where am I mistaken?
