Arman Darbinyan ; Rostislav Grigorchuk ; Asif Shaikh - Finitely generated subgroups of free groups as formal languages and their cogrowth

gcc:7617 - journal of Groups, complexity, cryptology, November 16, 2021, Volume 13, Issue 2 - https://doi.org/10.46298/jgcc.2021.13.2.7617
Finitely generated subgroups of free groups as formal languages and their cogrowthArticle

Authors: Arman Darbinyan ; Rostislav Grigorchuk ORCID; Asif Shaikh

    For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.


    Volume: Volume 13, Issue 2
    Published on: November 16, 2021
    Accepted on: November 8, 2021
    Submitted on: June 23, 2021
    Keywords: Mathematics - Group Theory,20E07, 68Q45, 68Q70

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