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  Finitely generated subgroups of free groups as formal languages and their cogrowth

Darbinyan, A., Grigorchuk, R., & Shaikh, A. (2021). Finitely generated subgroups of free groups as formal languages and their cogrowth. Journal of groups, complexity, cryptology, 13(2): Paper No. 1. doi:10.46298/jgcc.2021.13.2.7617.

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Item Permalink: https://hdl.handle.net/21.11116/0000-0009-FE5B-4 Version Permalink: https://hdl.handle.net/21.11116/0000-000C-3E7B-6
Genre: Journal Article

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Darbinyan-Grigorchuk-Shaikh_Finitely generated subgroups of free groups as formal languages and their cogrowth_2021.pdf (Publisher version), 512KB
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© A. Darbinyan, R. Grigorchuk and A. Shaikh This work is licensed under the Creative Commons Attribution License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or Eisenacher Strasse 2, 10777 Berlin, Germany
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https://doi.org/10.46298/jgcc.2021.13.2.7617 (Publisher version)
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 Creators:
Darbinyan, Arman, Author
Grigorchuk, Rostislav1, Author           
Shaikh, Asif, Author
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Group Theory
 Abstract: 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 $H$.

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Language(s): eng - English
 Dates: 2021
 Publication Status: Published online
 Pages: 30
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 2106.11552
DOI: 10.46298/jgcc.2021.13.2.7617
 Degree: -

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Title: Journal of groups, complexity, cryptology
Source Genre: Journal
 Creator(s):
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Publ. Info: EDP Sciences
Pages: - Volume / Issue: 13 (2) Sequence Number: Paper No. 1 Start / End Page: - Identifier: -