2024/07/23 by Shaikh, Asif
#20E05 #20F65 #20F69 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2407.16523
This article focuses on free factors H≤ Fm of the free group Fm with finite rank m > 2, and specifically addresses the implications of Ascari's refinement of the Whitehead automorphism φ for H as introduced in \citeascari2021fine. Ascari showed that if the core ΔH of H has more than one vertex, then the core Δφ(H) of φ(H) can be derived from ΔH. We consider the regular language LH of reduced words from Fm representing elements of H, and employ the construction of BH described in \citeDGS2021. BH is a finite ergodic, deterministic automaton that recognizes LH. Extending Ascari's result, we show that for the aforementioned free factors H of Fm, the automaton Bφ(H) can be obtained from BH. Further, we present a method for deriving the adjacency matrix of the transition graph of Bφ(H) from that of BH and establish that αH < αφ(H), where αH, αφ(H) represent the cogrowths of H and φ(H), respectively, with respect to a fixed basis X of Fm. The proof is based on the Perron-Frobenius theory for non-negative matrices.