Groups generated by 3-state automata over a 2-letter alphabet. II

Abstract

An approach to a classification of groups generated by 3-state automata over a 2-letter alphabet and the current progress in this direction are presented. Several results related to the whole class are formulated. In particular, all finite, abelian, and free groups are classified. In addition, we provide detailed information and complete proofs for several groups from the class, with the intention of showing the main methods and techniques used in the classification.

Automata generating free products of groups of order 2

“…There is an algorithm to check spherical transitivity of elements of self-similar groups acting on binary tree described explicitly in [26] (where a more general algorithm is given), [18] and implemented in [27]. But the method used in the above proof is a nice shortcut allowing to verify spherical transitivity much easier in the case when the factor of a self-similar group (acting on a binary tree) by the stabilizer of the second level is the cyclic group of order 4.…”

“…Since then, it is known as the Bellaterra automaton. It was proved by Muntyan (see the proof in [18] or [17]) that B 3 generates a group isomorphic to the free product of 3 copies of groups of order 2.…”

“…We cannot miss mentioning the amusing fact that among all 190 types of 3-state automata acting on the 2-letter alphabet, that are not symmetric to each other the only automaton generating a free non-abelian group is the first Aleshin automaton (see [18]). Thus pinpointing by Aleshin this unique automaton a quarter of a century ago should really be highly valued.…”

On the automaton complexity of wreath powers of non-abelian finite simple groups

“…Which groups are automata groups? Presently, there is an impressive family of such groups, including free groups of finite rank, some free products of finite cyclic groups, Baumslag-Solitar groups, lamplighter groups and other (see [2,4,6,10,12,14]). …”

Automaton representation of a free group

“…Such automata constitute a quite small part of all automata. Moreover, if we consider groups generated by initial subautomata of an automaton, then, as shown in [6], there exist 122 nonisomorphic groups generated by the automata with three states, and only the automaton described in [3] generates a free group. In [5], an automaton torsion-free group (not free) with three generating semigroup automata is given.…”