Absolutely Wild Automorphisms

• Absolutely wild automorphisms are a subclass of free algebra automorphisms that cannot be approximated by tame maps, exhibiting intrinsic nontameness at the infinitesimal level.
• They are rigorously detected using tangent Lie algebra techniques where the divergence of the associated derivation serves as a clear computable indicator.
• This refined concept enhances structural classification and algorithmic testing in automorphism groups, with implications spanning free algebras, matrix algebras, and Lie theoretic varieties.

Absolutely wild automorphisms are a refined and robust subclass of wild automorphisms in the theory of automorphism groups of free algebras and related algebraic structures. While wild automorphisms are those that cannot be decomposed into products of elementary or tame automorphisms, absolutely wild automorphisms are characterized by being intrinsically nontame even under approximations by tame maps—persisting as "maximally wild" even at the infinitesimal (tangent) level. The concept is rigorously developed via tangent Lie algebras and approximation theory, and distinguishes between wild automorphisms that arise as power series limits of tame automorphisms (approximately tame) and those that cannot (absolutely wild). This notion plays a crucial role in the structural and cohomological analysis of automorphism groups across several varieties of free algebras and highlights deep connections with derivation theory, divergence, and the geometry of group actions.

1. Definitions: Tame, Wild, Approximately Tame, and Absolutely Wild Automorphisms

Let $A = K_{\mathcal{M}}\langle X \rangle$ be a finitely generated free algebra in a variety $\mathcal{M}$, graded by degree, with associated automorphism group $\mathrm{Aut}(A)$. The standard definitions are as follows:

• Tame Automorphism: An automorphism is tame if it can be written as a finite product of elementary automorphisms (maps modifying a single variable by adding an element not involving that variable), possibly together with affine or linear automorphisms (depending on the structure of $A$).
• Wild Automorphism: An automorphism is wild if it is not tame, i.e., cannot be decomposed in this way.
• Approximately Tame Automorphism: $\varphi$ is approximately tame if there exists, for each $k$, a tame automorphism $\psi_k$ such that $\varphi \psi_k^{-1} \in \mathrm{IA}(k)$, where $\mathrm{IA}(k)$ is the subgroup of automorphisms acting identically mod the terms of degree $>k$ (the "t-adic" or power series topology). Thus, $\varphi$ can be approximated arbitrarily closely by tame maps.
• Absolutely Wild Automorphism (Editor's term): $\varphi$ is absolutely wild if it is not approximately tame; that is, no such approximation by tame automorphisms exists, regardless of the filtration degree. This means even the infinitesimal or tangent behavior of the automorphism is not captured by the tame subgroup (Shestakov et al., 28 Jul 2025).

The essential distinguishing feature is that absolutely wild automorphisms remain outside the closure of the tame subgroup with respect to the natural filtration on automorphisms, often reflecting a deeper rigidity in their tangent (first-order) structure.

2. Tangent Lie Algebras and the Detection of Absolute Wildness

A major technical advance is the use of tangent Lie algebras of automorphism groups for the detection of absolute wildness (Shestakov et al., 28 Jul 2025). For a filtered automorphism group $H \subseteq \mathrm{Aut}(A)$, the tangent Lie algebra $T(H)$ is formed by considering, for each $$\varphi \in \mathrm{IA}(i) \setminus \mathrm{IA}(i+1)$$, its deviation from identity modulo higher-degree terms:

$$\varphi = (x_1 + f_1 + F_1, \ldots, x_n + f_n + F_n)$$

with each $f_j$ homogeneous of degree $i+1$, and $F_j$ in higher degrees. Assign to $\varphi$ the derivation

$$T(\varphi) = f_1 \partial_{x_1} + \cdots + f_n \partial_{x_n} \in \mathfrak{L}_i \subseteq \mathrm{Der}(A)$$

where $\mathrm{Der}(A)$ is the Lie algebra of derivations graded by degree. The tangent Lie algebra $T(H)$ is then constructed as the graded subalgebra generated by such $T(\varphi)$. This structure captures the "infinitesimal" action of automorphisms.

A key observation is that, for many classical varieties (including Nielsen–Schreier varieties, free associative algebras, commutative algebras, and metabelian Lie algebras), $T(\mathrm{Aut}(A))$ is contained in the divergence-free subalgebra of $\mathrm{Der}(A)$; i.e., the subalgebra of derivations with constant divergence (modulo appropriate commutator/ideal structure) (Shestakov et al., 28 Jul 2025).

If a wild automorphism $\varphi$ induces a tangent derivation $T(\varphi)$ with nonzero divergence, then $\varphi$ cannot be approximated by tame automorphisms; thus, $\varphi$ is absolutely wild. Formally, for $D = D_F$ with $F = (f_1, \ldots, f_n)$,

$$\operatorname{div}(D) = f_1/x_1 + \cdots + f_n/x_n \pmod{[\mathcal{U}, \mathcal{U}] + R}$$

where $\mathcal{U}$ is the universal enveloping algebra of $A$ and $R$ is the Jacobson radical. Nonzero divergence modulo these relations gives a computable criterion for absolute wildness (Shestakov et al., 28 Jul 2025).

3. Paradigmatic Examples and Contrast to Classical Wild Automorphisms

The distinction between absolutely wild automorphisms and merely wild (but approximately tame) automorphisms is sharp. Prominent examples include:

Automorphism	Variety	Wildness Type	Tame Approximation?
Bergman automorphism	Free matrix algebra	Absolutely wild	No
Nagata automorphism	Polynomial algebra, 3 variables	Wild, not absolutely wild	Yes
Anick automorphism	Free associative algebra	Conjecturally wild, not absolutely wild	Yes (in known cases)
Many exponential automorphisms built from non-triangulable locally nilpotent derivations	Nonassociative free algebras (Novikov, Lie varieties)	Absolutely wild	No

For the Bergman automorphism of the free matrix algebra of order two,

$\varphi = (x_1 + [x_1, x_2]^2, x_2)$

the tangent derivation $T(\varphi)$ has nonzero divergence, confirming absolute wildness (Shestakov et al., 28 Jul 2025).

By contrast, the Nagata automorphism and Anick automorphism, while wild by the criteria of Shestakov–Umirbaev and others, can be approximated arbitrarily closely by tame automorphisms in the power series topology, so they are not absolutely wild (Shestakov et al., 28 Jul 2025). This underlines that the set of absolutely wild automorphisms is a proper subset of the wild automorphisms.

4. Absolute Wildness in Classical and Nonclassical Varieties

Extending this framework, the paper (Shestakov et al., 28 Jul 2025) shows that free algebras in any polynilpotent variety of Lie algebras—except for abelian and metabelian cases—possess absolutely wild automorphisms. The detection is based on tangent derivations with nonzero divergence, as polynilpotent Lie structures admit enough freedom to construct such derivations that cannot be generated by the tangent algebra of tame automorphisms.

For Nielsen–Schreier-type varieties, the tangent algebra generated by tame automorphisms coincides with that of the entire automorphism group, so every automorphism is approximately tame. In contrast, in other varieties (such as free nonassociative algebras, matrix algebras, and specific Poisson or Novikov algebras), the divergence computation via the tangent Lie algebra detects absolute wildness.

5. Distinction from Tame Closure and Topological Considerations

Absolutely wild automorphisms are sensitive to the closure of the tame subgroup in the formal topology on automorphism groups. In dimensions two, by the Jung–van der Kulk theorem and analogous results, all automorphisms are tame and this subtlety does not arise. For polynomial rings in three or more variables, free associative algebras, or more generally more complex varieties, the group of tame automorphisms is not closed in the ind-topology (Edo et al., 2014), and thus wild automorphisms may be approximated by sequences of tame automorphisms. Only those wild automorphisms that cannot even be so approximated are absolutely wild.

This distinction is crucial, since certain wild automorphisms—such as degenerations of tame automorphisms—arise as limits of tame maps, but absolutely wild automorphisms do not exist in the topological closure of the tame subgroup. The detection is thus of both algebraic and topological nature.

6. Implications, Applications, and Further Directions

• Structural Classification: The concept of absolute wildness offers a refined stratification of automorphism groups in algebraic and Lie-theoretic settings, important for understanding the extent and boundaries of algorithmic and combinatorial reduction methods.
• Algorithmics: Criteria based on tangent Lie algebra and divergence permit concrete, often algorithmic tests for absolute wildness—by computing the divergence of the tangent derivation associated to a map (Shestakov et al., 28 Jul 2025).
• Generalization Potential: The methodology points toward a program for classifying automorphism groups in varieties beyond associative and commutative algebras, with current results encompassing many Nielsen–Schreier varieties, matrix algebras, Poisson algebras, and various classes of Lie and Novikov algebras.
• Open Problems: Among the notable exceptions to the absolute wildness criterion are the Nagata and Anick automorphisms. Whether these are the only such exceptions remains open; verifying absolute wildness for further wild automorphisms—both classical and in new settings—constitutes a central direction.
• Extension to Other Invariants: The tangent Lie algebra construction is adaptable to settings involving Fox derivatives, Jacobians, and other invariants. For instance, in metabelian and polynilpotent cases, relations between tangent derivations and the Jacobian determine wildness/tameness properties (Umirbaev, 14 Jan 2024).

7. Summary Formulas and Characterizations

• Definition of Approximate Tameness:

$$\forall k, \exists \psi_k \text{ tame}, \quad \varphi \psi_k^{-1} \in \mathrm{IA}(k)$$

• Absolute Wildness:

$$\varphi \text{ is absolutely wild} \iff \forall (\psi_k) \text{ tame},\, \exists k:\ \varphi \psi_k^{-1} \notin \mathrm{IA}(k)$$

• Tangent Derivation:

$$T(\varphi) = f_1 \partial_1 + \cdots + f_n \partial_n$$

• Divergence Criterion:

$$\operatorname{div}(T(\varphi)) = \text{Tr}(J(T(\varphi))) \pmod{[\mathcal{U},\mathcal{U}] + R}$$

If $\operatorname{div}(T(\varphi)) \neq 0$, then $\varphi$ is absolutely wild.

By isolating absolutely wild automorphisms via tangent Lie algebra techniques—in particular, through the divergence of their tangent derivations—the framework supports a general and computable methodology to distinguish layers within wild automorphism theory. This has implications for the structure, dynamics, and computability in the theory of algebraic and noncommutative automorphism groups (Shestakov et al., 28 Jul 2025).