Non-Homogeneous Automorphisms in Algebra and Geometry

• Non-homogeneous automorphisms are structure-preserving maps that do not align with canonical gradings, impacting classification in algebra and geometry.
• They are constructed in diverse settings—such as affine toric varieties, Grassmann algebras, and Riemann surface covers—demonstrating varied algebraic and geometric behavior.
• Studying these automorphisms reveals novel group structures and rigidity phenomena, driving advances in understanding symmetry and classification of mathematical objects.

Non-homogeneous automorphisms are automorphisms of algebraic, geometric, or combinatorial structures that do not preserve a canonical or standard grading, symmetry, or orbit decomposition. The term “non-homogeneous” distinguishes such automorphisms from those arising from group actions with open dense or transitive orbits, or those compatible with a pre-imposed grading. Recent research has clarified both their construction in various contexts and their impact on the structure and classification of automorphism groups.

1. Definitions and General Framework

Non-homogeneous automorphisms arise in contexts where the automorphism group of a structure contains elements that do not align with any obvious grading or homogeneous action. This can happen in algebraic structures such as the Grassmann algebra (Guimarães, 21 Aug 2025), Riemann surface covers (Hidalgo, 7 Jul 2024), affine Cremona groups (&&&2&&&), toric varieties (Díaz et al., 2022), and affine surfaces (Kovalenko, 2013, Aitzhanova et al., 2022).

Key situations include:

• Automorphisms not compatible with a group action yielding an open dense orbit (almost homogeneous varieties (Brion, 2018)).
• Automorphisms or gradings not derived from eigenvectors or “pure” elements (Grassmann algebra).
• Lifts of automorphisms in covering spaces that do not produce split exact sequences (Riemann surfaces).
• Polynomial or birational automorphisms in varieties generated by derivations or reversions not arising from group-theoretic or grading constraints.

Canonical contrasting examples:

• Homogeneous automorphisms: preserve grading or act transitively (e.g., affine Veronese surfaces (Aitzhanova et al., 2022), canonical grading in the Grassmann algebra).
• Non-homogeneous automorphisms: fail to split, mix orbit structures, or involve generators not mapping to ± themselves.

2. Construction in Algebraic and Geometric Settings

2.1 Affine and Toric Varieties

In affine toric geometry (Díaz et al., 2022), automorphism groups are often constructed from normalized additive group actions (i.e., 𝔾ₐ-actions) that correspond to homogeneous locally nilpotent derivations. Extension to non-normal varieties shows that the group Aut(X) can be isomorphic to that of a normal variety, with non-homogeneous automorphisms not arising from the torus action but still tightly controlled via generalizations of Demazure roots:

$\partial_{α}(\chi^m) = \rho(m) \cdot \chi^{m+α}$

where α is a Demazure root and ρ is a distinguished ray.

2.2 Grassmann Algebra

The Grassmann algebra exhibits a rich supply of non-homogeneous automorphisms of order 2 (Guimarães, 21 Aug 2025). Construction typically involves:

• Selecting an infinite subset I of indices, with a partition into $I^{+}, I^{-}$ for which $φ(e_i) = \pm e_i$.
• For $j \notin I$, setting $φ(e_j) = -e_j + d_j$ with $d_j$ constructed to meet the necessary anti-commutativity and order-2 conditions.

Type 2, 3, and 4 automorphisms introduce various correction terms and modifications:

• Type 3: Finitely many modified generators, e.g.,

$φ(e_n) = -e_n + \lambda_n (e_1 \cdots e_{k+t}) e_n$

• Type 4: No generator is an eigenvector for φ; explicit combinatorial constructions ensure $φ^2(e_i) = e_i$ but $φ(e_i) \neq \pm e_i$.

2.3 Non-homogeneous Gizatullin Surfaces

Automorphisms constructed via birational maps obtained from paired reversions at points in large sets of C*-orbits span a free subgroup (Kovalenko, 2013). The freeness is certified by a minimal-length decomposition and isolation from algebraic subgroups:

• For $a \in A$, define $P_a = v_a \circ v_{o(a)}^{-1}$, yielding a free group $F$ with trivial intersection with Aut(V)_alg.

2.4 Homology Covers of Riemann Surfaces

Given a Riemann surface S with a non-abelian fundamental group, the k-homology cover $\widetilde{S}_k$ has a deck transformation group $M_k \cong H_1(S; \mathbb{Z}_k)$ (Hidalgo, 7 Jul 2024). For $L \leq Aut(S)$, the lifted automorphism group $\widetilde{L}_k$ fits into the exact sequence:

$1 \to M_k \to \widetilde{L}_k \to L \to 1$

If this sequence does not split, automorphisms of $\widetilde{S}_k$ cannot be separated into direct products of $L$ and $M_k$, physically manifesting non-homogeneous automorphisms.

3. Rigidity Results and Classification

3.1 Cremona Groups and Tame/Wild Automorphisms

The affine Cremona group Aut($\mathbb{C}^n$) contains both tame (homogeneous) and wild (non-homogeneous) automorphisms. A key result (Stampfli, 2012) is that any abstract automorphism θ that fixes a closed torus (and hence the tame subgroup) must also fix large classes of non-tame automorphisms, including the Nagata automorphism:

$$N = \exp(pD), \quad D = -2y(\partial/\partial x) + z (\partial/\partial y), \quad p = xz + y^2$$

If $f \in \ker D \setminus \mathbb{C}[z]$, $\exp(fD)$ is non-tame and θ fixes all such automorphisms whenever it fixes the tame subgroup.

3.2 Quasi-homogeneity from Contracting Automorphisms

Any complex analytic singularity admitting a contracting automorphism must be quasi-homogeneous (Morvan, 16 Dec 2024). The proof involves embedding the singularity in $\mathbb{C}^d$, conjugating the automorphism to a Poincaré-Dulac normal form, and demonstrating that the invariant ideal is generated by weighted homogeneous polynomials.

4. Impact on Group Structure and Presentations

4.1 Automorphism Groups as Free or Amalgamated Products

Research shows that automorphism groups generated by non-homogeneous automorphisms can have highly nontrivial structure:

• For smooth Gizatullin surfaces, the subgroup $F$ generated by paired reversions is free and its intersection with Aut(V)_alg is trivial (Kovalenko, 2013).
• The automorphism group of a Veronese surface’s coordinate ring admits an amalgamated free product structure, controlled by graded automorphisms (Aitzhanova et al., 2022).

4.2 Arithmetic Properties and Infinite Presentations

Almost homogeneous varieties exhibit automorphism groups with linear/algebraic and arithmetic structures, with the possibility of infinite component groups (Brion, 2018). Open questions remain as to full classification, realizability of all linear algebraic groups as automorphism groups, and structure in positive characteristic.

For Higman-Thompson groups and their overgroups, presentations and subgroup structures of automorphism-like groups are open for investigation (Bleak et al., 2016).

5. Broader Applications and Open Problems

Non-homogeneous automorphisms play critical roles in:

• Determining rigidity properties of symmetry groups,
• Classifying affine and projective varieties,
• Studying moduli of Riemann surfaces and covers,
• Understanding dynamical consequences and gradings of operator algebras,
• Extending the scope of automorphism group presentations and the theory of infinite-dimensional transformation groups.

Representative open problems include:

• Classifying all non-homogeneous automorphisms up to graded isomorphism (conjecture for Grassmann algebras: all $\mathbb{Z}_2$-gradings are isomorphic to homogeneous gradings (Guimarães, 21 Aug 2025)).
• Establishing criteria for splitting/non-splitting of exact sequences in lifted automorphism groups (Hidalgo, 7 Jul 2024).
• Extending the rigidity results for ind-groups, Cremona groups, and toric varieties to higher dimensions and broader categories.

Comparative table: Non-homogeneous automorphism contexts

Structure	Non-homogeneous automorphism	Classification/Impact
Grassmann algebra (E)	φ not ±id on all generators	Conjectured isomorphic to homogeneous
Gizatullin/affine surfaces (V)	Free group by reversions	Not generated by algebraic subgroups
Toric varieties (X)	Lifts not torus-normalized	Aut(X) ≅ Aut(normalization)
Cremona groups Aut(ℂ³)	Modifications exp(fD)	Fixed by automorphisms fixing tame
Riemann surface covers Sₖ	Non-split exact sequences	Extensions not semidirect products

6. Conclusion

Non-homogeneous automorphisms highlight subtle and often rigid features of algebraic, geometric, and combinatorial objects. Across multiple settings, they influence group structure, connect with rigidity phenomena, and pose significant classification problems. The interplay between homogeneity, grading, and automorphism types remains central to ongoing research in geometry, group theory, and algebraic dynamics.