An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

Lewis, Drew; Perry, Kaitlyn; Straub, Armin

doi:10.1016/j.jpaa.2019.04.002

Cited by 4 publications

(2 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Answering our question in Remark 5.7.2, D. Lewis, K. Perry, and A. Straub proved recently (see [35,Cor. 21]) that for n = 2 the group generated by the root subgroups H 1 = {(x, y) ↦ (x + t 1 y 2 , y)} and H 2 = {(x, y) ↦ (x, y + t 2 x)}, t 1 , t 2 ∈ K acts infinitely transitively on A 2 ∖ 0.…”

Section: 19mentioning

confidence: 73%

Infinite transitivity, finite generation, and Demazure roots

Arzhantsev

Kuyumzhiyan

Zaidenberg

2019

Advances in Mathematics

View full text Add to dashboard Cite

An affine algebraic variety X of dimension ≥ 2 is called flexible if the subgroup SAut(X) ⊂ Aut(X) generated by the one-parameter unipotent subgroups acts m-transitively on reg (X) for any m ≥ 1. In [4] we proved that any nondegenerate toric affine variety X is flexible. In the present paper we show that one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property, provided the toric variety X is smooth in codimension two. For X = A n with n ≥ 2, three such subgroups suffice. G a -subgroup, for short. One can consider the subgroup SAut(X) ⊂ Aut(X) generated by all the one-parameter unipotent subgroups. It is known ([4, Thm. 2.1]) that for a toric affine variety X with no torus factor (that is, a nondegenerate toric variety) the group SAut(X) acts infinitely transitively on the smooth locus reg(X), that is, m-transitively for any m ≥ 1. Varieties X with this property are called flexible ([3, 4]). Actually, the simple transitivity of

show abstract

Section: 19mentioning

confidence: 73%

Infinite transitivity, finite generation, and Demazure roots

Arzhantsev

Kuyumzhiyan

Zaidenberg

2019

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…1.3]; such subgroups are found explicitly in [And19]. For instance, for n = 2 the group G generated by two root subgroups U 1 = {(x, y) ↦ (x + t 1 y 2 , y)} and U 2 = {(x, y) ↦ (x, y + t 2 x)}, t 1 , t 2 ∈ k acts highly transitively on A 2 ∖ {0} equipped with the standard action of the 2-torus, see [LPS18,Cor. 21].…”

Section: Introductionmentioning

confidence: 99%

Tits type alternative for groups acting on toric affine varieties

Arzhantsev¹,

Zaidenberg²

2020

Preprint

View full text Add to dashboard Cite

Given a toric affine algebraic variety X and a collection of one-parameter unipotent subgroups U 1 , . . . , U s of Aut(X) which are normalized by the torus acting on X, we show that the group G generated by U 1 , . . . , U s verifies the Tits alternative, and, moreover, either is a unipotent algebraic group, or contains a nonabelian free subgroup. We deduce that if G is m-transitive for any positive integer m, then G contains a nonabelian free subgroup, and so, is of exponential growth.

show abstract

Automorphisms of algebraic varieties and infinite transitivity

Arzhantsev¹

2023

St. Petersburg Math. J.

View full text Add to dashboard Cite

This is a survey of recent results on multiple transitivity for automorphism groups of affine algebraic varieties. The property of infinite transitivity of the special automorphism group is treated, which is equivalent to the flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of varieties. Also, the situations are studied where infinite transitivity occurs for automorphism groups generated by finitely many one-parameter subgroups. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are discussed.

show abstract

An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

Cited by 4 publications

References 17 publications

Infinite transitivity, finite generation, and Demazure roots

Infinite transitivity, finite generation, and Demazure roots

Tits type alternative for groups acting on toric affine varieties

Automorphisms of algebraic varieties and infinite transitivity

Contact Info

Product

Resources

About