The cogrowth inequality from Whitehead's algorithm

TL;DR

This paper investigates how Ascari's refinement of Whitehead's algorithm affects the automaton recognizing free factors in free groups, demonstrating that the cogrowth decreases under the automorphism using Perron-Frobenius theory.

Contribution

It extends Ascari's results by showing how automata recognizing free factors transform under automorphisms and proves the decrease in cogrowth via matrix spectral analysis.

Findings

Automaton for phi(H) can be derived from that of H.

The adjacency matrix of the automaton transforms predictably under automorphisms.

Cogrowth decreases when applying the automorphism phi.

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.