Tame Automorphisms Research Articles

On the acylindrical hyperbolicity of the tame automorphism group of SL2(C)

We introduce the notion of \"uber-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of non-simplicity, of groups acting on non locally compact spaces of arbitrary dimension. We also give a simple local criterion to construct \"uber-contracting elements for groups acting on complexes with unbounded links of vertices. As an application, we show the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$, a subgroup of the $3$-dimensional Cremona group, through its action on a CAT(0) square complex recently introduced by Bisi-Furter-Lamy.

A Lifting of an Automorphism of a K3 Surface over Odd Characteristic

In this paper, we prove that, over an algebraically closed field of odd characteristic, a weakly tame automorphism of a K3 surface of finite height can be lifted over the ring of Witt vectors of the base field. Also we prove that a non-symplectic tame automorphism of a supersingular K3 surface or a symplectic tame automorphism of a supersingular K3 surface of Artin-invariant at least 2 can be lifted over the ring of Witt vectors. Using these results, we prove, for a weakly tame K3 surface of finite height, there is a lifting over the ring of Witt vectors to which whole the automorphism group of the K3 surface can be lifted. Also we prove a K3 surface equipped with a purely non-symplectic automorphism of a certain order is unique up to isomorphism.

On tameness of Matsumoto-Imai central maps in three variables over the finite field $\mathbb F_2$

Triangular transformation method (TTM) is one of the multivariate public key cryptosystems (MPKC) based on the intractability of tame decomposition problem. In TTM, a special class of tame automorphisms are used to construct encryption schemes. However, because of the specificity of such tame automorphisms, it is important to evaluate the computational complexity of the tame decomposition problem for secure use of MPKC. In this paper, as the first step for security evaluations, we focus on Matsumoto-Imai cryptosystems. We shall prove that the Matsumoto-Imai central maps in three variables over $\mathbb{F}_{2}$ is tame, and we describe the tame decompositions of the Matsumoto-Imai central maps.

Degeneration of Tame Automorphisms of a Polynomial Ring

Recently, Edo and Poloni constructed a family of tame automorphisms of a polynomial ring in three variables which degenerates to a wild automorphism. In this note, we generalize the example by a different method.

Tame Automorphisms with Multidegrees in the Form of Arithmetic Progressions

Abstract Let (a, a + d, a + 2d) be an arithmetic progression of positive integers. The following statements are proved: (1) If a | 2d, then (a, a + d, a + 2d) ∈ mdeg(Tame(ℂ3)). (2) If a ∤ 2d and (a, a + d, a + 2d) ∉ {(4i, 5i, 6i), (4i, 7i, 10i) : i ∈ ℕ+}, then (a, a + d, a + 2d) ∉ mdeg(Tame(ℂ3)).

The affine automorphism group of A3 is not a maximal subgroup of the tame automorphism group

The affine automorphism group of A3 is not a maximal subgroup of the tame automorphism group

The profinite polynomial automorphism group

We introduce an extension of the (tame) polynomial automorphism group over finite fields: the profinite (tame) polynomial automorphism group, which is obtained by putting a natural topology on the automorphism group. We show that most known candidate non-tame automorphisms are inside the profinite tame polynomial automorphism group, giving another result showing that tame maps are potentially “dense” inside the set of automorphisms. We study the profinite tame automorphism group and show that it is not far from the set of bijections obtained by endomorphisms.

On Tame and Wild Automorphisms of Algebras

Let Bn be a polynomial algebra of n variables over a field F. Considering a free associative algebra An of rank n over F as a polynomial algebra of noncommuting variables, we choose the ideal R of all polynomials with a zero absolute term in Bn and An. The well-known concept of wild automorphisms of the algebras An and Bn is transferred to R; the study of wild automorphisms is reduced to monic automorphisms of the algebra R, i.e., those identical on each factor Rk/Rk+1. In particular, this enables us to study the properties of the known Nagata and Anik automorphisms in detail. For n = 3 we investigate the hypothesis that the Anik automorphism is tame modulo Rk for every given integer k > 1.

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M n ) of a free metabelian Lie algebra M n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M 4) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M 3) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.

On the Closure of the Tame Automorphism Group of Affine Three-Space

We provide explicit families of tame automorphisms of the complex affine three-space which degenerate to wild automorphisms. This shows that the tame subgroup of the group of polynomial automorphisms of C3 is not closed, when the latter is seen as an infinite-dimensional algebraic group.

On the Karaś type theorems for the multidegrees of polynomial automorphisms

To solve Nagata's conjecture, Shestakov–Umirbaev constructed a theory for deciding wildness of polynomial automorphisms in three variables. Recently, Karaś and others studied multidegrees of polynomial automorphisms as an application of this theory. They give various necessary conditions for triples of positive integers to be multidegrees of tame automorphisms in three variables. In this paper, we prove a strong theorem unifying and generalizing these results using the generalized Shestakov–Umirbaev theory.

The tame automorphism group of an affine quadric threefold acting on a square complex

We study the group Tame(SL 2 ) of tame automorphisms of a smooth affine 3-dimensional quadric, which we can view as the underlying variety of SL 2 (ℂ). We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT(0) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame(SL 2 ) is linearizable, and that Tame(SL 2 ) satisfies the Tits alternative.

Tame Automorphisms of Elementary Free Groups

We prove that the automorphism group of a finitely generated fully residually free group is tame.

Coordinates of R[x,y]: Constructions and Classifications

Let R be a PID. We construct and classify all coordinates of R[x, y] of the form p 2 y + Q 2(p 1 x + Q 1(y)) with p 1, p 2 ∈ qt(R) and Q 1, Q 2 ∈ qt(R)[y]. From this construction (with R = K[z]) we obtain nontame automorphisms σ of K[x, y, z] (where K is a field of characteristic 0) such that the subgroup generated by σ and the affine automorphisms contains all tame automorphisms.

Multidegrees of tame automorphisms with one prime number

Let $3\leq d_1\leq d_2\leq d_3$ be integers. We show the following results: (1) If $d_2$ is a prime number and $\frac{d_1}{\gcd(d_1,d_3)}\neq2$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_1=d_2$ or $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$; (2) If $d_3$ is a prime number and $\gcd(d_1,d_2)=1$, then $(d_1,d_2,d_3)$ is a multidegree of a tame automorphism if and only if $d_3\in d_1\mathbb{N}+d_2\mathbb{N}$. We also relate this investigation with a conjecture of Drensky and Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials, and we give a counter-example to this conjecture.

Separability of wild automorphisms of a polynomial ring

For a large class (including the Nagata automorphism) of wild automorphisms F of k[x, y, z] (where k is a field of characteristic zero), we prove that we can find a weight w such that there exists no tame automorphism with the same w-weight multidegree.

On Automorphisms of the Affine Cremona Group

We show that every automorphism of the group 𝒢 n :=Aut(𝔸 n ) of polynomial automorphisms of complex affine n-space 𝔸 n =ℂ n is inner up to field automorphisms when restricted to the subgroup T𝒢 n of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension n=2 where all automorphisms are tame: T𝒢 2 =𝒢 2 . The methods are different, based on arguments from algebraic group actions.

Almost Tame Automorphisms of Two Generated Free Leibniz Algebras

Denote a free Leibniz algebra in two variables over a field K by LB ⟨ x, y ⟩. The groups of almost triangular AT(LB ⟨ x, y ⟩) and almost tame automorphisms Tame(LB ⟨ x, y ⟩) of LB ⟨ x, y ⟩ are introduced here. These groups extend the groups of triangular and tame automorphisms, respectively. We prove that where H = GL 2(K) ∩ AT(LB ⟨ x, y ⟩). We also show that Tame(LB ⟨ x, y ⟩) is smaller than the group of automorphisms of LB ⟨ x, y ⟩.

Endomorphisms of polynomial algebras with small co-invariants

Let ϕ be an endomorphism of the polynomial algebra C[x1,…,xn] and let co(ϕ) be the dimension of the smallest sub-coalgebra C(ϕ) of the Hopf algebra C[Gan] containing ϕ(x1),…,ϕ(xn). We show that ϕ is a tame automorphism if detJ(ϕ)∈C∗ and one of the following conditions is satisfied: (i) co(ϕ)≤n+2; (ii) ϕ is quadratic and co(ϕ)≤n+5.

Groups of triangular automorphisms of a free associative algebra and a polynomial algebra

We study the structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semidirect product of abelian groups. Using this decomposition we describe the structure of the lower central series and the derived series for the group of unitriangular automorphisms and prove that every element of the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rank greater than 2 is not linear and describe some two-generated subgroups of this group. Also we give a more simple system of generators for the group of tame automorphisms in comparison to that system in Umirbaevʼs paper.