Mixed-identity-freeness and primitivity of group rings

Technical Summary: Mixed-identity-freeness and Primitivity of Group Rings

Problem Statement
The paper addresses the longstanding problem of determining conditions under which a group ring $RG$ (or group algebra $KG$) is primitive. A ring is (right) primitive if it possesses a faithful irreducible (right) module. While significant progress has been made for specific classes of groups—such as polycyclic-by-finite groups (Domanov, Farkas–Passman, Roseblade), free products (Formanek, Balogun, Alexander–Nishinaka), and hyperbolic/acylindrically hyperbolic groups (Solie, Abbott–Dahmani)—a unified algebraic criterion applicable to a broader spectrum of non-noetherian groups was lacking. The specific goal is to establish primitivity for group rings over domains $R$ where $|R| \le |G|$, and over any field $K$, for a wider class of countable groups.

Methodology
The author introduces and leverages the concept of mixed-identity-free (MIF) groups to derive a new algebraic criterion for primitivity. The methodology proceeds in two main stages:

1. Algebraic Criterion via Property $(\ast)$:
   The paper utilizes a property $(\ast)$ introduced by Alexander and Nishinaka, which guarantees primitivity if satisfied by a group containing a non-abelian free subgroup of the same cardinality. Property $(\ast)$ requires that for any finite set of non-trivial elements $M \subset G \setminus \{1\}$ and integer $m \ge 2$, there exist distinct elements $a, b, c \in G$ such that any product of the form $\prod_{i=1}^m (x_i^{-1} g_i x_i) = 1$ (with $g_i \in M$ and $x_i \in \{a, b, c\}$) forces a repetition in the conjugating elements ($x_i = x_{i+1}$ for some $i$).
   The core technical contribution is proving that MIF groups satisfy property $(\ast)$. This is achieved by:

   • Constructing specific commutator words $W_j$ recursively to show that MIF groups allow for the simultaneous non-trivial evaluation of finitely many non-trivial words in $G * F_n$.
   • Adapting "big powers" lemmas to construct an element $u \in G$ of order at least 4 such that specific "mixed" words involving powers of $u$ and elements from $M$ are non-trivial.
   • Using this element $u$ to define the distinct conjugating elements $a=u, b=u^2, c=u^3$, thereby satisfying property $(\ast)$.

2. Dynamical Criterion:
   The paper establishes a link between algebraic properties and dynamical systems. It defines an action $G \curvearrowright X$ as extremely proximal (if for any non-empty open $U, V$, there exists $g$ such that $g(X \setminus U) \subseteq V$) and topologically free (if fixed points of non-identity elements are nowhere dense).
   The author proves that if a countable group admits a topologically free, extremely proximal action, it is necessarily MIF and contains a non-abelian free subgroup. This connects the algebraic result to dynamical properties often found in groups acting on trees or Cantor sets.

Key Contributions and Results

• Theorem 1.2: If $G$ is a MIF group containing a non-abelian free subgroup of cardinality $|G|$, then $RG$ is primitive for any domain $R$ with $|R| \le |G|$, and $KG$ is primitive for any field $K$.
• Theorem 1.4: If a countable group $G$ admits a topologically free, extremely proximal action on a Hausdorff space $X$ ($|X| > 2$), then $RG$ is primitive for any countable domain $R$, and $KG$ is primitive for any field $K$.
• Recovery of Existing Results: The new criterion provides an independent proof for the primitivity of group rings for:
  • All acylindrically hyperbolic groups without finite normal subgroups (recovering Abbott–Dahmani).
  • All non-elementary hyperbolic groups and non-elementary free products (recovering Solie and Formanek).
• Expansion to New Examples: The criterion positively applies to a "plethora" of groups previously not covered by these specific primitivity results, including:
  • Thompson-like groups (Higman–Thompson $G_{n,r}$, Brin–Thompson $nV$, Röver's group).
  • Commensurator groups of non-elementary torsion-free hyperbolic groups.
  • Certain Kac–Moody groups and Burger–Mozes groups.
  • Groups acting on trees and certain linear groups.

Significance
The paper claims significance by unifying the study of primitivity for group rings under the umbrella of mixed-identity-freeness. By demonstrating that MIF is a sufficient condition for primitivity (given the presence of a non-abelian free subgroup), the work extends known results to a vast class of groups that are often studied for their dynamical properties (e.g., extremely proximal actions) rather than purely algebraic ones. The dynamical criterion (Theorem 1.4) is particularly notable as it allows for the verification of primitivity through the geometric or topological behavior of group actions, a method that is likely to yield further examples as the study of extremely proximal actions expands in the context of $C^*$-algebras and geometric group theory. The author notes that this approach recovers major existing theorems while simultaneously opening the door to proving primitivity for many new, complex groups.