The development of the theory of automatic groups
automatic group, hyperbolic group, finite state automaton, combing, 3-manifold group, decision problem, word problem, conjugacy problem, Artin group, Coxeter group, mapping class group.
Status to appear in `In the tradition of Thurston II', K. Oshika and A. Papadopolous (editors), Springer 2022
We describe the development of the theory of automatic groups.
We begin with a historical introduction, define the concepts
of automatic, biautomatic and combable groups,
derive basic properties, then explain how hyperbolic groups and the groups
of compact 3-manifolds based on six of Thurston's eight geometries can be
We describe software developed in Warwick to compute automatic structures, as
well as the development of practical algorithms that use those structures.
We explain how actions of groups on spaces displaying various notions of
negative curvature can be used to prove automaticity or biautomaticity,
and show how these results have been used to derive these properties for
groups in some infinite families (braid groups, mapping class groups, families of Artin groups, and Coxeter groups).
Throughout the text we flag up open problems as well as problems that
remained open for some time but have now been resolved.
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