Triangulable subgroups of the Cremona groups

The talk is aimed to elaborate on the Bass' Triangulability Problem about unipotent algebraic subgroups of the affine Cremona groups. The following topics are discussed: a triangulability criterion, the existence of non-triangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range the Bass' Triangulability Problem is answered in the affirmative. These results are based on a general theorem about invariant subfields of purely transcendental field extensions of transcendental degree 1. A general construction of all rationally triangulable subgroups of the Cremona groups is given, and, as an application, all rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups are classified up to conjugacy.