The cogrowth inequality from Whitehead's algorithm
Abstract
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 phi for H as introduced in ascari2021fine. Ascari showed that if the core DeltaH of H has more than one vertex, then the core Deltaphi(H) of phi(H) can be derived from DeltaH. We consider the regular language LH of reduced words from Fm representing elements of H, and employ the construction of mathcalBH described in DGS2021. mathcalBH 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 mathcalBphi(H) can be obtained from mathcalBH. Further, we present a method for deriving the adjacency matrix of the transition graph of mathcalBphi(H) from that of mathcalBH and establish that alphaH < alphaphi(H), where alphaH, alphaphi(H)$ represent the cogrowths of H and phi(H), respectively, with respect to a fixed basis X of Fm. The proof is based on the Perron-Frobenius theory for non-negative matrices.
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