Subgroups of the Cremona groups: Bass’ problem
A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rational triangulability. The latter property answers in the affirmative Bass’ Trian gulability Problem in the stable range. A general construction of all rationally triangulable connected solvable affine algebraic subgroups of the Cremona groups is obtained. As an application, a classification of all ratio nally triangulable connected onedimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy is given.