October 2, 20250 citationsOpen Access

Borel subgroups of the automorphism groups of affine toric surfaces

IAIvan ArzhantsevNational Research University Higher School of Economics MZMikhail ZaidenbergCentre National de la Recherche Scientifique

Key Points

Borel subgroups are identified as defining features of automorphism groups of affine toric surfaces.
The cyclic quotients of the affine plane are categorized into two groups based on the behavior of their Borel subgroups.
The study employs the bass-serre-tits theory to understand groups acting on trees and their properties.
Understanding these structures aids in theoretical explorations of algebraic geometry and related fields.

Abstract

In I. Arzhantsev and M. Zaidenberg, Acyclic curves and group actions on affine toric surfaces. Affine Algebraic Geometry, 1--41. World Scientific Publishing Co. 2013 we described the automorphism groups of the cyclic quotients of the affine plane. In this article, we study the Borel subgroups and, more generally, the maximal solvable subgroups of these ind-groups. We show that the cyclic quotients of the affine plane are divided into two species. In one of them, the Borel subgroups form a single conjugacy class, while in the other, there are two conjugacy classes of Borel subgroups. The proofs explore the Bass-Serre-Tits theory of groups acting on trees.

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