### Book chapter

## Bass' triangulability problem

Exploring Bass’ Triangulability Problem on unipotent algebraic

subgroups of the affine Cremona groups, we prove a triangulability

criterion, the existence of nontriangulable connected solvable affine algebraic

subgroups of the Cremona groups, and stable triangulability

of such subgroups; in particular, in the stable range we answer Bass’

Triangulability Problem in the affirmative. To this end we prove a theorem

on invariant subfields of 1-extensions. We also obtain a general

construction of all rationally triangulable subgroups of the Cremona

groups and, as an application, classify rationally triangulable connected

one-dimensional unipotent affine algebraic subgroups of the Cremona

groups up to conjugacy.