  The Cremona group is the group of birational transformations of the complex
projective plane. In this paper we classify its subgroups that consist only of
elliptic elements using elementary model theory. This yields in particular a
description of the structure of torsion subgroups. As an appliction, we prove
the Tits alternative for arbitrary subgroups of the Cremona group, generalizing
a result of Cantat. We also describe solvable subgroups of the Cremona group
and their derived length, refining results from D\'eserti.
