Stable Primitivity Rank

• Stable primitivity rank is a stabilized invariant that refines classical primitivity rank by measuring the persistent failure of elements in free groups to be primitive.
• It employs diagrammatic methods, such as Stallings core graphs and covering diagrams, to filter out trivial algebraic behaviors and yield a quantifiable measure.
• Its applications span representation theory, operator algebras, and tensor theory, linking algebraic topology, combinatorics, and random measures.

The stable primitivity rank is an invariant arising in group theory and algebraic topology, designed to measure, in a stabilized sense, the failure of an element (or tuple) to be primitive: that is, not contained in a free generating set or not satisfy a prescribed algebraic independence. It is defined by refining the classical notion of primitivity rank such that, even under stabilization or passage to larger structures, the essential obstruction to primitivity persists and remains quantifiable. This concept has now been investigated in the context of free groups, group representations, C*-algebras, higher-rank graphs, tensor theory, and word measures on groups, with profound implications for topology, combinatorics, random walks, and representation theory.

1. Classical Primitivity Rank and Its Stabilization

For a free group $F_r$ of rank $r \geq 2$, the primitivity rank $\pi(w)$ of a nontrivial word $w \in F_r$ is the minimal rank of a subgroup $H \leq F_r$ such that $w \in H$ and $w$ is not primitive in $H$ (Kapovich, 2021). If no such subgroup exists (i.e., $w$ is primitive in $F_r$), one sets $\pi(w)=\infty$. The critical set $\operatorname{Crit}(w)$ consists of all rank-$\pi(w)$ subgroups containing $w$ in which $w$ is not primitive.

However, this notion can be unstable under algebraic operations such as taking powers, extensions, or stabilizations. The stable primitivity rank, denoted $s(w)$, is introduced to "smooth out" these effects, typically by considering all nontrivial stabilizations (via coverings, powers, or redundant generators) and discounting trivial algebraic phenomena. For example, the naïve limit $\lim_n \pi(w^n)/n$ is non-informative, as proper powers may artificially lower the minimal witnessing rank. The definition via effective covering diagrams (see §2) ensures only genuine complexity is measured (Puder et al., 2023).

2. Diagrammatic and Topological Definitions

The formalism for stable primitivity rank uses Stallings core graphs and topological covering diagrams. Given $w \in F_r$, consider the bouquet $\Omega$ of $r$ circles ($\pi_1(\Omega) \cong F_r$). Every subgroup $H \leq F_r$ determines a unique immersed finite core graph $\Gamma_H \to \Omega$. A commutative diagram

$$P \xrightarrow{b} \Gamma \xrightarrow{f} \Omega \ \downarrow \rho \hspace{4em} \downarrow \ \mathbb{S}^1 \xrightarrow{w} \Omega$$

is constructed, where $P$ is a disjoint union of circles, $\rho$ is a finite-degree covering map, $b$ is an immersion, and $f$ is a core immersion. Efficiency conditions are imposed: $b$ should not be an isomorphism on any component, ensuring only "genuine" stabilization is captured.

The stable primitivity rank is then defined as

$$s(w) = \inf\left\{ -\chi(\Gamma) / \deg(\rho) : \text{(diagram as above, with efficiency)} \right\},$$

where $\chi(\Gamma)$ is the Euler characteristic and $\deg(\rho)$ is the covering degree (Puder et al., 2023). This stable invariant generalizes the combinatorial primitivity rank by discounting trivial cases (e.g., $s(w)=0$ for proper powers) and permits rational values.

3. Stable Primitivity Rank in Representation Theory and Topology

In the context of representations of free groups into Lie groups, a closely related notion arises: a representation $\rho : F_n \to PSL(2,\mathbb{C})$ (or more generally into semisimple Lie groups) is primitive stable if all axes of primitive elements map, under the orbit map, to uniformly quasigeodesic curves in hyperbolic space (Minsky et al., 2010, Kim et al., 2015). This property can, perhaps surprisingly, persist under arbitrary stabilization: by adding redundant generators to the domain free group, the representation can remain primitive stable, even as the rank $n \to \infty$, while the geometric type of the image group (and its quotient manifold) remains fixed (Minsky et al., 2010). The stable primitivity rank of a representation is thus the minimal rank to which the domain free group can be stabilized with primitive stability preserved.

Analytically, the Whitehead graph furnishes a criterion: for a cyclically reduced word $[w]$ and a generating set $X$, its Whitehead graph $\mathrm{Wh}([w], X)$ is constructed, and primitive stability relates to its connectivity. If the graph is "cut-point-free," primitive stability persists. This topological viewpoint connects stable primitivity rank to Heegaard splittings, knot complements, and flypes (operations increasing genus with stability preserved) (Minsky et al., 2010).

4. Stable Primitivity Rank in Operator Algebras

The concept of "stable rank" in C*-algebra theory, as developed by Rieffel, is a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its stable rank $\operatorname{sr}(A)$ is the minimal $n$ for which the set of $n$-tuples generating $A$ densely in norm is nonempty (Farah et al., 2016, Pask et al., 2020). The stable primitivity rank is, by plausible analogy, the minimal rank under stabilization such that the algebra remains stably finite and retains desired regularity properties, e.g., density of invertibles or faithful irreducible representations.

Techniques from logic of metric structures show the stable rank is axiomatizable and continuous under ultraproducts and Kadison–Kastler perturbations. This suggests that a logical definition of stable primitivity rank would inherit invariance under such perturbations and ultraproducts. Structural results for graph C*-algebras tie the stable rank (and thus primitivity phenomena) to combinatorial features: for higher-rank graphs, absence of cycles with entrances and the ranks of periodicity groups predict stable rank and, by extension, stable primitivity behaviors (Pask et al., 2020).

5. Connections to Stable Invariants and Word Measures

The theory of stable primitivity rank is one facet of a broader suite of "stable invariants" (including stable commutator length $\operatorname{scl}(w)$ and stable square length $\operatorname{ssql}(w)$) that control the asymptotics of word measures on compact and finite groups (Puder et al., 2023). For a given word $w$, the expected value of stable irreducible characters under the induced word measure on symmetric groups $S_N$ satisfies

$$\mathbb{E}_w[\chi^{\mu[N]}] = \Theta\left((\dim \chi^{\mu[N]})^{-s(w)}\right),$$

where $s(w)$ governs the decay exponent. For unitary groups, similar asymptotic laws are determined by other stable invariants (e.g., $2 \operatorname{scl}(w)$).

A key conjecture in the literature is that for non-power words,

$s(w) = \pi(w) - 1,$

with $s(w)$ a 'degree-1' invariant (see (Puder et al., 2023); conjecture due to Wilton). Moreover, $s(w)$ is a profinite invariant: if two words induce identical word measures on all finite groups, their stable primitivity ranks coincide.

6. Stable Primitivity Rank in Random Graphs and Group Elements

There is a probabilistic dimension: for random elements in free groups, the primitivity rank $\pi(w)$ stably attains its maximum possible value—namely, for a generic subset $V \subset F_r$, all $w \in V$ satisfy $\pi(w) = r$ and $\operatorname{Crit}(w) = \{F_r\}$ (Kapovich, 2021). This stability, persisting even as the word is subjected to random walks or generic constructions, indicates that for large classes of elements, the stable primitivity rank reflects maximal algebraic complexity.

Algorithmic procedures (such as Whitehead's algorithm and analysis of cyclically reduced forms via covering graphs) reveal that critical sets can be effectively computed and that stability phenomena are generic with respect to exponential density in the group.

7. Stable Primitivity Rank in Tensor and Field Extension Theory

Recent work extends stability properties to tensor ranks over finite fields (Moshkovitz et al., 5 Nov 2024). The analytic rank of a tensor is shown to be stable under field extensions, with constants independent of the base field. This uniformity was achieved by bounding classical rank and subrank of the multiplication tensor, independent of field size, via function field towers. A plausible implication is that if the primitivity rank of tensors can be formulated analogously, one should expect similar uniform stability over field extensions. This situates stable primitivity rank as one among several tensor invariants behaving uniformly under extensions.

Summary Table: Stable Primitivity Rank Across Domains

Context	Definition/Key Characteristic	Stability Phenomenon
Free groups (words/elements)	$\pi(w)$: minimal rank, $s(w)$: stabilized via diagrams	$s(w)$ attains max value generically
Representations (PSL(2,C), $\mathrm{PGL}(m,\mathbb{R})$)	Minimal domain rank for primitive-stable representations	Arbitrary stabilization allowed
C*-algebras (stable rank)	Minimal size of generating tuple for stable finiteness	Invariance under ultraproducts, KK
Graph C*-algebras	Determined by cycles/periodicity of underlying graph	Explicit stable rank formulas
Tensors over finite fields	Analytic, geometric, primitivity ranks	Uniform stability over extensions
Word measures on groups	Controls decay of stable character expectations	Profinite invariant

Concluding Remarks

The stable primitivity rank offers a robust invariant across mathematical disciplines for quantifying stabilized non-primitivity, with definitions amenable to topological, combinatorial, and algebraic formalization. Its stability properties—often uniform under algebraic operations, perturbations, and extensions—make it a central tool for analyzing representation-theoretic measures, operator algebra regularity, geometric parameters of group actions, and combinatorial features in random and structured group elements. Conjectural equalities between stable primitivity rank and classical ranks (modulo stabilizing corrections) link it with a family of stable invariants currently active in frontier research, indicating deep interrelations between group theory, low-dimensional topology, noncommutative geometry, and random matrix theory.