The cogrowth inequality from Whitehead's algorithm
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912853650309120 |
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| author | Shaikh, Asif |
| author_facet | Shaikh, Asif |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16523 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The cogrowth inequality from Whitehead's algorithm Shaikh, Asif Group Theory 20E05, 20F69, 20F65 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. |
| title | The cogrowth inequality from Whitehead's algorithm |
| topic | Group Theory 20E05, 20F69, 20F65 |
| url | https://arxiv.org/abs/2407.16523 |