The cogrowth inequality from Whitehead's algorithm
Abstract
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.
Cite
@article{arxiv.2407.16523,
title = {The cogrowth inequality from Whitehead's algorithm},
author = {Asif Shaikh},
journal= {arXiv preprint arXiv:2407.16523},
year = {2026}
}
Comments
18 pages, 4 figures. A detailed introduction has been added. Corollary 4.1 has been modified. Minor revisions were made throughout to improve the exposition