Closed Subgroups of the Polynomial Automorphism Group Containing the Affine Subgroup

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

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“…. , l + 1 verify (18) and provide the desired decomposition. This lemma and Theorem 5.14 lead to the following corollary.…”

Section: 1mentioning

confidence: 99%

Infinite transitivity, finite generation, and Demazure roots

Arzhantsev

Kuyumzhiyan

Zaidenberg

2019

Advances in Mathematics

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“…We used Sage combined with Magma (and, independently, Mathematica; though not in all cases) to verify the two hypotheses of Lemma 12 and establish the following results: 6) .…”

Section: Computer-aided Resultsmentioning

confidence: 99%

“…We note that these include the previously unknown special cases G (8) ⊂ G (5,4) , G (10) ⊂ G (6,5) , and G (12) ⊂ G (7,6) (among others). These would be quite tedious to establish directly (with currently known techniques), as the Gröbner bases involved in the computations can consist of a large number of polynomials, depending on the term order chosen.…”

Section: Computer-aided Resultsmentioning

confidence: 99%

An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

Lewis

Perry

Straub

2019

Journal of Pure and Applied Algebra

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We study the interaction between two structures on the group of polynomial automorphisms of the affine plane: its structure as an amalgamated free product and as an infinite-dimensional algebraic variety. We introduce a new conjecture, and show how it implies the Polydegree Conjecture. As the new conjecture is an ideal membership question, this shows that the Polydegree Conjecture is algorithmically decidable. We further describe how this approach provides a unified and shorter method of recovering existing results of Edo and Furter.

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“…Besides the above examples and the triangular subgroup B 2 , the only other maximal closed subgroup of Aut(A 2 C ) that we are aware of is the affine subgroup A 2 . The fact that A 2 is maximal among all closed subgroups of Aut(A 2 C ) is a particular case of the following recent result of Edo [Edo16]. (We recall that the so-called tame subgroup of Aut(A 2 C ) is its subgroup generated by A n and B n .)…”

Section: Maximal Closed Subgroupsmentioning

confidence: 92%

On the maximality of the triangular subgroup

Furter¹,

Poloni²

2016

Preprint

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We prove that the subgroup of triangular automorphisms of the complex affine n-space is maximal among all solvable subgroups of Aut(A n C ) for every n. In particular, it is a Borel subgroup of Aut(A n C ), when the latter is viewed as an indgroup. In dimension two, we prove that the triangular subgroup is a maximal closed subgroup and that nevertheless, it is not maximal among all subgroups of Aut(A 2 C ). Given an automorphism f of A 2 C , we study the question whether the group generated by f and the triangular subgroup is equal to the whole group Aut(A 2 C ).

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Closed Subgroups of the Polynomial Automorphism Group Containing the Affine Subgroup

Abstract: We prove that, in characteristic zero, closed subgroups of the polynomial automorphisms group containing the affine group contain the whole tame group.

Cited by 4 publications

References 9 publications

Infinite transitivity, finite generation, and Demazure roots

Infinite transitivity, finite generation, and Demazure roots

An algorithmic approach to the Polydegree Conjecture for plane polynomial automorphisms

On the maximality of the triangular subgroup

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