Automatic group

In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata represent the Cayley graph of the group, i. e. can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator.

More precisely, let G be a group and A be a finite set of generators. Then an automatic structure of G with respect to A is a set of finite-state automata:

• the word-acceptor, which accepts for every element of G at least one word in $A^\ast$ representing it
• multipliers, one for each $a \in A \cup \{1\}$, which accept a pair (w₁, w₂), for words w_i accepted by the word-acceptor, precisely when $w_1 a = w_2$ in G.

The property of being automatic does not depend on the set of generators.

The concept of automatic groups generalizes naturally to automatic semigroups.

Properties

Automatic groups have word problem solvable in quadratic time. More strongly, a given word can actually be put into canonical form in quadratic time, based on which the word problem may be solved by testing whether the canonical forms of two words are equal.

Examples of automatic groups

The automatic groups include:

• Finite groups. To see this take the regular language to be the set of all words in the finite group.
• Euclidean groups
• All finitely generated Coxeter groups
• Geometrically finite groups

Examples of non-automatic groups

• Baumslag-Solitar groups
• Non-Euclidean nilpotent groups

Biautomatic groups

A group is biautomatic if it has two multiplier automata, for left and right multiplication by elements of the generating set respectively. A biautomatic group is clearly automatic.

Examples include:

• Hyperbolic groups.
• Any Artin group of finite type including braid groups.

Automatic structures

The idea of describing algebraic structures with finite-automata can be generalized from groups to other structures.

Additional reading

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• Epstein et al. (1992), Section 2.3, "Automatic Groups: Definition", pp. 45–51.
• Epstein et al. (1992), Section 2.4, "Invariance under Change of Generators", pp. 52–55.
• Epstein et al. (1992), Section 6.1, "Semigroups and Specialized Axioms", pp. 114–116.
• Epstein et al. (1992), Theorem 2.3.10, p. 50.