Bass' triangulability problem

TL;DR

This paper investigates Bass' Triangulability Problem within the context of unipotent algebraic subgroups of the affine Cremona groups, establishing criteria, existence results, and classifications related to triangulability.

Contribution

It introduces a triangulability criterion, demonstrates the existence of nontriangulable subgroups, and classifies rationally triangulable one-dimensional unipotent subgroups.

Findings

Established a criterion for triangulability.

Proved the existence of nontriangulable subgroups.

Classified rationally triangulable one-dimensional unipotent subgroups.

Abstract

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 is 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.