163
       J     o     u     r     n     a       l       A       l     g      e       b     r     a        D       i     s     c     r     e      t     e        M     a      t       h  . Algebra and Discrete Mathematics RESEARCH AR TICLE Number 1.  (2008).  pp. 1 – 163 c  Journal “Algebra and Discrete Mathematics” On classication of groups generated by 3-state automata over a 2-letter alphabet Ievgen Bondarenko, Rostislav Grigorchuk, Rostyslav Kravchenko, Yevgen Muntyan, Volodymyr Nekrashevych, Dmytro Savchuk and Zoran  ˇ Suni ´ c Dedicated to V. V. Kirichenko on his 65th birthday and V. I. Sushchansky on his 60th birthday Abstract.  W e show tha t the class of gro ups generated by 3-state automata over a 2-letter alphabet has no more than 122 mem bers. For each group in the class we provide some basic in- formation, 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 innite order (we show that no group in the class is an innite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary cal- cul ati ons . F or some of the example s, we establish that the y are (virtually) iterated monodromy groups of post-critically nite ra- tional functions, in which cases we describe the functions and the limit spaces. The re are exa ctl y 6 ni te gr oups in the cla ss (of or- der no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelia n group (of rank 3). The other exam- ples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups  BS (1, ±3), and a free product  C 2  ∗ C 2  ∗ C 2 ) to enticing (Basilica group and a few other iterated monodromy groups). All authors were partially supported by at least one of the NSF grants DMS-308985, DMS-456185, DMS-600975, and DMS-605019 2000 Mathematics Subject Classication:  20E08. Key words and phrases:  automata groups, self-similar groups, branch groups.

Algebraic Combinatorics

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