The cogrowth inequality from Whitehead's algorithm

This article focuses on free factors H <= F_m of the free group F_m with finite rank m > 2, and specifically addresses the implications of Ascari’s refinement of the Whitehead automorphism phi for H as introduced in \cite{ascari2021fine}. Ascari showed that if the core Delta_H of H has more than one vertex, then the core Delta_{phi(H)} of phi(H) can be derived from Delta_H. We consider the regular language L_H of reduced words from F_m representing elements of H, and employ the construction of mathcal{B}H described in \cite{DGS2021}. mathcal{B}H is a finite ergodic, deterministic automaton that recognizes L_H. Extending Ascari’s result, we show that for the aforementioned free factors H of F_m, the automaton mathcal{B}{phi(H)} can be obtained from mathcal{B}H. Further, we present a method for deriving the adjacency matrix of the transition graph of mathcal{B}{phi(H)} from that of mathcal{B}H and establish that alpha_H < alpha{phi(H)}, where alpha_H, alpha{phi(H)}$ represent the cogrowths of H and phi(H), respectively, with respect to a fixed basis X of F_m. The proof is based on the Perron-Frobenius theory for non-negative matrices.

💡 Research Summary

The paper investigates the relationship between the cogrowth of a non‑cyclic free factor H of a free group Fₘ (m > 2) and the cogrowth of its image under a Whitehead automorphism φ that arises from Ascari’s refinement of Whitehead’s algorithm. The authors begin by recalling Ascari’s theorem: if the core graph Δ_H of H has more than one vertex, there exists a Whitehead automorphism φ such that the core graph Δ_{φ(H)} can be obtained from Δ_H by collapsing a specific set of edges, thereby strictly reducing both the number of vertices and edges.

The central object of study is the regular language L_H consisting of reduced words in Fₘ that represent elements of H. Following the construction in Dicks‑Gould‑Sullivan (2021), the authors use the finite, ergodic, deterministic automaton B_H to recognize L_H. B_H is derived from the extended core graph 𝔅Δ_H, which contains both forward and backward edges of the Schreier graph of H. The automaton’s state set consists of pairs (vertex, label) that correspond to admissible transitions in 𝔅Δ_H, and it is shown to be minimal, strongly connected, deterministic, and to have homogeneous ambiguity equal to deg(v₁) − 1, where v₁ is the distinguished root vertex.

A key technical bridge is the adjacency matrix M_H of the transition digraph of B_H. Because B_H is ergodic, M_H is a non‑negative, integral, irreducible matrix. By Perron‑Frobenius theory, M_H possesses a unique maximal eigenvalue λ_H > 0, and the entropy of the language satisfies ent(L_H) = log λ_H. The cogrowth α_H of H (defined as the lim sup of the n‑th root of the number a_n of reduced words of length n in H) coincides with λ_H: α_H = e^{ent(L_H)} = λ_H.

The authors then turn to the Whitehead automorphism φ = (A, a), defined by a distinguished letter a ∈ Σ and a subset A ⊂ Σ \ {a, a⁻¹}. The automorphism fixes a and transforms each generator x_j according to its membership in A and the presence of its inverse, yielding the classic four‑case definition. The Whitehead graph of a cyclically reduced word is introduced to explain how cut vertices signal possible length‑reducing moves.

The main contributions are two theorems. Theorem C (Theorem 4.1) proves that the automaton B_{φ(H)} recognizing L_{φ(H)} can be obtained from B_H by collapsing precisely the same edges that Ascari’s graph reduction collapses. In matrix terms, the adjacency matrix M_{φ(H)} is derived from M_H by adding a non‑negative perturbation ΔM that reflects the merged transitions; the resulting matrix remains irreducible.

Theorem D (Theorem 4.3) applies Perron‑Frobenius theory to show λ_H < λ_{φ(H)}. The proof observes that the edge‑collapsing operation strictly increases at least one entry of the matrix while preserving non‑negativity, which forces the spectral radius to increase. Consequently, α_H = λ_H < λ_{φ(H)} = α_{φ(H)}. This inequality demonstrates that the Whitehead reduction, when lifted to the language‑theoretic setting of automata, not only simplifies the core graph but also strictly improves the cogrowth (or, equivalently, the exponential growth rate of reduced words) of the subgroup.

The paper concludes by posing the “Whitehead maximal cogrowth problem”: given a finitely generated subgroup H ≤ Fₘ, determine an automorphism φ in the Whitehead orbit of H that maximizes α_{φ(H)}. The authors note that no effective algorithm is currently known, and they suggest that further exploration of Ascari’s edge‑collapsing technique or cut‑capacity methods might yield progress. An illustrative example and a list of open questions round out the discussion, indicating promising directions for future research at the intersection of combinatorial group theory, formal language theory, and spectral graph theory.