Groups defined by automata

📝 Abstract

This is Chapter 24 in the “AutoMathA” handbook. Finite automata have been used effectively in recent years to define infinite groups. The two main lines of research have as their most representative objects the class of automatic groups (including word-hyperbolic groups as a particular case) and automata groups (singled out among the more general self-similar groups). The first approach implements in the language of automata some tight constraints on the geometry of the group’s Cayley graph, building strange, beautiful bridges between far-off domains. Automata are used to define a normal form for group elements, and to monitor the fundamental group operations. The second approach features groups acting in a finitely constrained manner on a regular rooted tree. Automata define sequential permutations of the tree, and represent the group elements themselves. The choice of particular classes of automata has often provided groups with exotic behaviour which have revolutioned our perception of infinite finitely generated groups.

💡 Analysis

This is Chapter 24 in the “AutoMathA” handbook. Finite automata have been used effectively in recent years to define infinite groups. The two main lines of research have as their most representative objects the class of automatic groups (including word-hyperbolic groups as a particular case) and automata groups (singled out among the more general self-similar groups). The first approach implements in the language of automata some tight constraints on the geometry of the group’s Cayley graph, building strange, beautiful bridges between far-off domains. Automata are used to define a normal form for group elements, and to monitor the fundamental group operations. The second approach features groups acting in a finitely constrained manner on a regular rooted tree. Automata define sequential permutations of the tree, and represent the group elements themselves. The choice of particular classes of automata has often provided groups with exotic behaviour which have revolutioned our perception of infinite finitely generated groups.

📄 Content

Groups defined by automata Laurent Bartholdi1 Pedro V. Silva2,∗ 1 Mathematisches Institut Georg-August Universit¨at zu G¨ottingen Bunsenstraße 3–5 D-37073 G¨ottingen, Germany email: laurent.bartholdi@gmail.com 2 Centro de Matem´atica, Faculdade de Ciˆencias Universidade do Porto R. Campo Alegre 687 4169-007 Porto, Portugal email: pvsilva@fc.up.pt 2010 Mathematics Subject Classification: 20F65, 20E08, 20F10, 20F67, 68Q45 Key words: Automatic groups, word-hyperbolic groups, self-similar groups. Contents 1 The geometry of the Cayley graph 102 1.1 History of geometric group theory . . . . . . . . . . . . . . . . . . . . . 103 1.2 Automatic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 1.3 Main examples of automatic groups . . . . . . . . . . . . . . . . . . . . 109 1.4 Properties of automatic groups . . . . . . . . . . . . . . . . . . . . . . . 110 1.5 Word-hyperbolic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 1.6 Non-automatic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2 Groups generated by automata 115 2.1 Main examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.2 Decision problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.3 Bounded and contracting automata . . . . . . . . . . . . . . . . . . . . . 121 2.4 Branch groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.5 Growth of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.6 Dynamics and subdivision rules . . . . . . . . . . . . . . . . . . . . . . . 125 2.7 Reversible actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.8 Bireversible actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 References 132 ∗The second author acknowledges support by Project ASA (PTDC/MAT/65481/2006) and C.M.U.P., fi- nanced by F.C.T. (Portugal) through the programmes POCTI and POSI, with national and E.U. structural funds. arXiv:1012.1531v1 [cs.FL] 7 Dec 2010 102 L. Bartholdi, P. V. Silva Finite automata have been used effectively in recent years to define infinite groups. The two main lines of research have as their most representative objects the class of automatic groups (including “word-hyperbolic groups” as a particular case) and automata groups (singled out among the more general “self-similar groups”). The first approach is studied in Section 1 and implements in the language of automata some tight constraints on the geometry of the group’s Cayley graph. Automata are used to define a normal form for group elements and to execute the fundamental group operations. The second approach is developed in Section 2 and focuses on groups acting in a finitely constrained manner on a regular rooted tree. The automata define sequential per- mutations of the tree, and can even represent the group elements themselves. The authors are grateful to Martin R. Bridson, Franc¸ois Dahmani, Rostislav I. Grig- orchuk, Luc Guyot, and Mark V. Sapir for their remarks on a preliminary version of this text. 1 The geometry of the Cayley graph Since its inception at the beginning of the 19th century, group theory has been recognized as a powerful language to capture symmetries of mathematical objects: crystals in the early 19th century, for Hessel and Frankenheim [53, page 120]; roots of a polynomial, for Galois and Abel; solutions of a differential equation, for Lie, Painlev´e, etc. It was only later, mainly through the work of Klein and Poincar´e, that the tight connections between group theory and geometry were brought to light. Topology and group theory are related as follows. Consider a space X, on which a group G acts freely: for every g ̸= 1 ∈G and x ∈X, we have x · g ̸= x. If the quotient space Z = X/G is compact, then G “looks very much like” X, in the following sense: choose any x ∈X, and consider the orbit x · G. This identifies G with a roughly evenly distributed subset of X. Conversely, consider a “nice” compact space Z with fundamental group G: then X = eZ, the universal cover of Z, admits a free G-action. In conclusion, properties of the fundamental group of a compact space Z reflect geometric properties of the space’s universal cover. We recall that finitely generated groups were defined in §23.1: they are groups G admitting a surjective map π : FA ↠G, where FA is the free group on a finite set A. Definition 1.1. A group G is finitely presented if it is finitely generated, say by π : FA ↠ G, and if there exists a finite subset R ⊂FA such the kernel ker(π) is generated by the FA-conjugates of R, that is, ker(π) = ⟨⟨R⟩⟩; one then has G = FA/⟨⟨R⟩⟩. These r ∈R are called relators of the presentation; and one writes G = ⟨A | R⟩. Sometimes it is convenient to write a relator in the form ‘a = b’ rather than the more exact form ‘ab−1’. Let G be a finitely generated group, with generating set A. Its Cayley graph C (G, A), Groups defined by automata 103 introduced by Cayley [44], is the graph with vertex set G a

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