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On classification of groups generated by 3-state automata over a 2-letter alphabet

On classification of groups generated by 3-state automata over a 2-letter alphabet

We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients...

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Main Authors: Bondarenko, I., Grigorchuk, R., Kravchenko, R., Muntyan, Y., Nekrashevych, V., Savchuk, D., Sunic, Z.
Format: Article
Language:English
Published: Інститут прикладної математики і механіки НАН України 2008
Series:Algebra and Discrete Mathematics
Online Access:http://dspace.nbuv.gov.ua/handle/123456789/152389
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spelling irk-123456789-1523892019-06-11T01:25:42Z On classification of groups generated by 3-state automata over a 2-letter alphabet Bondarenko, I. Grigorchuk, R. Kravchenko, R. Muntyan, Y. Nekrashevych, V. Savchuk, D. Sunic, Z. We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients by level stabilizers (congruence quotients), and hystogram of the spectrum of the adjacency operator of the Schreier graph of the action on level 9. In most cases we provide more information, such as whether the group is contracting, self-replicating, or (weakly) branch group, and exhibit elements of infinite order (we show that no group in the class is an infinite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary calculations. For some of the examples, we establish that they are (virtually) iterated monodromy groups of post-critically finite rational functions, in which cases we describe the functions and the limit spaces. There are exactly 6 finite groups in the class (of order no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelian group (of rank 3). The other examples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups BS(1±3), and a free product C2 ∗ C2 ∗ C2) to enticing (Basilica group and a few other iterated monodromy groups). 2008 Article On classification of groups generated by 3-state automata over a 2-letter alphabet / I. Bondarenko, R. Grigorchuk, R. Kravchenko, Y. Muntyan, V. Nekrashevych, D. Savchuk, Z. Sunic // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 1. — С. 1–163. — Бібліогр.: 50 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20E08. http://dspace.nbuv.gov.ua/handle/123456789/152389 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients by level stabilizers (congruence quotients), and hystogram of the spectrum of the adjacency operator of the Schreier graph of the action on level 9. In most cases we provide more information, such as whether the group is contracting, self-replicating, or (weakly) branch group, and exhibit elements of infinite order (we show that no group in the class is an infinite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary calculations. For some of the examples, we establish that they are (virtually) iterated monodromy groups of post-critically finite rational functions, in which cases we describe the functions and the limit spaces. There are exactly 6 finite groups in the class (of order no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelian group (of rank 3). The other examples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups BS(1±3), and a free product C2 ∗ C2 ∗ C2) to enticing (Basilica group and a few other iterated monodromy groups).
format Article
author Bondarenko, I.
Grigorchuk, R.
Kravchenko, R.
Muntyan, Y.
Nekrashevych, V.
Savchuk, D.
Sunic, Z.
spellingShingle Bondarenko, I.
Grigorchuk, R.
Kravchenko, R.
Muntyan, Y.
Nekrashevych, V.
Savchuk, D.
Sunic, Z.
On classification of groups generated by 3-state automata over a 2-letter alphabet
Algebra and Discrete Mathematics
author_facet Bondarenko, I.
Grigorchuk, R.
Kravchenko, R.
Muntyan, Y.
Nekrashevych, V.
Savchuk, D.
Sunic, Z.
author_sort Bondarenko, I.
title On classification of groups generated by 3-state automata over a 2-letter alphabet
title_short On classification of groups generated by 3-state automata over a 2-letter alphabet
title_full On classification of groups generated by 3-state automata over a 2-letter alphabet
title_fullStr On classification of groups generated by 3-state automata over a 2-letter alphabet
title_full_unstemmed On classification of groups generated by 3-state automata over a 2-letter alphabet
title_sort on classification of groups generated by 3-state automata over a 2-letter alphabet
publisher Інститут прикладної математики і механіки НАН України
publishDate 2008
url http://dspace.nbuv.gov.ua/handle/123456789/152389
citation_txt On classification of groups generated by 3-state automata over a 2-letter alphabet / I. Bondarenko, R. Grigorchuk, R. Kravchenko, Y. Muntyan, V. Nekrashevych, D. Savchuk, Z. Sunic // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 1. — С. 1–163. — Бібліогр.: 50 назв. — англ.
series Algebra and Discrete Mathematics
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first_indexed 2023-05-20T17:38:16Z
last_indexed 2023-05-20T17:38:16Z
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