Ample generics in automorphism groups of Boolean powers of simple Mal'cev algebras

Problem Statement
The paper addresses the existence of "ample generics" in the automorphism groups of specific algebraic structures known as filtered Boolean powers. Let $A$ be a finite simple Mal'cev algebra (a class encompassing finite simple groups, modules, rings, associative and Lie algebras, loops, and quasigroups). The authors investigate the automorphism group of the filtered Boolean power $(A^{2^\omega})^x_e$, which consists of continuous functions from the Cantor space $2^\omega$ to $A$ that map specific distinct points $x = (x_1, \dots, x_n)$ to fixed singleton subalgebras $e = (e_1, \dots, e_n)$.

While ample generics (the existence of generic $n$-tuples for all $n$ under the diagonal conjugation action) are known for many automorphism groups of countable homogeneous relational structures and homeomorphism groups of topological spaces, their presence in the automorphism groups of algebraic structures has been less explored. The paper aims to prove that these automorphism groups possess ample generics, a property that implies significant topological and algebraic consequences such as the small index property, automatic continuity, and the Bergman property.

Methodology
The proof is bifurcated into two distinct cases based on the nature of the algebra $A$: the abelian case and the non-abelian case.

1. The Abelian Case:

   • Framework: The authors utilize the approach introduced by Kechris and Rosendal, which links ample generics to the model-theoretic properties of the class of finite structures.
   • Strategy: They characterize finite simple abelian Mal'cev algebras as reducts of finite simple modules. They demonstrate that the class of finite direct powers of such an algebra forms a Fraïssé class.
   • Key Steps: The proof establishes that the class of these finite structures, when expanded with partial isomorphisms, satisfies the Joint Embedding Property (JEP) and the Weak Amalgamation Property (WAP). A crucial simplification in this case is the Extension Property for Partial Automorphisms (EPPA), where every partial isomorphism on a direct power extends to a total automorphism. This allows the application of Kechris and Rosendal's theorem to conclude that the Fraïssé limit (which is isomorphic to the relevant Boolean power) has ample generics.

2. The Non-Abelian Case:

   • Obstacles: The standard Fraïssé approach fails here because the class of finite direct powers of a non-abelian simple Mal'cev algebra lacks the Hereditary Property and the EPPA. Furthermore, the Boolean power itself is not the Fraïssé limit of this class.
   • Decomposition Strategy: The authors employ a structural decomposition of the automorphism group $G = \text{Aut}((A^{2^\omega})^x_e)$. They show that $G$ splits as a semidirect product $K \rtimes H$, where:
     • $H$ is isomorphic to the pointwise stabilizer of the points $x$ in the homeomorphism group $\text{Homeo}_{2^\omega}$.
     • $K$ is a normal subgroup isomorphic to a closure of a filtered Boolean power of the automorphism group of $A$.
   • Intermediate Result (Theorem 1.2): Before tackling the full group, the authors prove that the pointwise stabilizer of finitely many points in $\text{Homeo}_{2^\omega}$ has ample generics. This generalizes a result by Kwiatkowska (who proved ample generics for $\text{Homeo}_{2^\omega}$ itself) using projective Fraïssé limits.
   • Combinatorial Core: To lift the genericity from $H$ to the semidirect product $G$, the authors develop a combinatorial argument involving "spiral structures" and "labellings." They prove that any translate of a generic tuple in $H$ by an element of $K$ can be expressed as a conjugate of that tuple by an element of $K$. This relies on Lemma 5.2, which establishes the existence of preimages for surjective structures satisfying a "quotient property" relative to group labellings.

Key Contributions and Results

• Theorem 1.1: The primary result establishes that for any finite simple Mal'cev algebra $A$, the automorphism group of the filtered Boolean power $(A^{2^\omega})^x_e$ has ample generics. This applies to a wide range of classical structures including groups, rings, and Lie algebras.
• Theorem 1.2: The paper proves that the pointwise stabilizer of finitely many points in the homeomorphism group of the Cantor space has ample generics. This extends Kwiatkowska's result and is a critical intermediate step for the non-abelian case.
• Corollary 1.3: As a consequence of the abelian case analysis, the paper shows that the general linear group $GL_V$, projective linear group $PGL_V$, and affine general linear group $AGL_V$ for a countable-dimensional vector space $V$ over a finite or countably infinite field have ample generics.
• Corollary 1.4 & 1.5: The existence of ample generics implies that these automorphism groups possess the small index property, automatic continuity, a unique Polish group topology, uncountable (strong) cofinality, and the Bergman property.

Significance
The paper claims to establish a new class of algebraic structures whose automorphism groups exhibit ample generics. Prior to this work, examples of such groups were largely restricted to automorphism groups of countable homogeneous relational structures or homeomorphism groups of topological spaces. The authors note that even for classical structures like finite simple groups or rings, this result is new.

By unifying the treatment of these diverse algebraic structures under the umbrella of Mal'cev algebras and Boolean powers, the paper provides a comprehensive proof that their automorphism groups share deep topological and algebraic properties (specifically the small index property and automatic continuity). The work bridges the gap between the theory of Boolean powers (a link between algebra and logic) and the model-theoretic study of generic automorphisms.

Open Questions
The paper concludes by noting an open question regarding free algebras. While the automorphism group of the filtered Boolean power $(A^{2^\omega})^x_1$ has the small index property, it remains unknown whether the automorphism group of the free algebra of countable rank in the variety generated by a finite simple non-abelian group (or Mal'cev algebra) possesses the small index property or ample generics.