The cogrowth inequality from Whitehead’s algorithm

Abstract

This article focuses on non-cyclic free factors $H\le F_m$ 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 Ascari (2024). 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 Darbinyan et al. (2021). The automaton ${\mathcal{B}}_H$ is finite, ergodic, deterministic, and 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.