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Automatic structures on central extensions. (English) Zbl 1114.20305
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14–19, 1996. Berlin: de Gruyter (ISBN 3-11-016366-7/hbk). 261-280 (1999).
Summary: We show that a central extension of a group $$H$$ by an Abelian group $$A$$ has an automatic structure with $$A$$ a rational subgroup if and only if $$H$$ has an automatic structure for which the extension is given by a “regular” cocycle. This had been proved for biautomatic structures by W. D. Neumann and L. Reeves [Int. J. Algebra Comput. 6, No. 3, 313-324 (1996; Zbl 0928.20028)]. We make a start at classifying automatic structures on such groups, but we show that, at least for automatic structures, a classification using “controlling subgroups”, as done by the authors in certain other cases, is impossible.
For the entire collection see [Zbl 0910.00040].
##### MSC:
 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F65 Geometric group theory 20J05 Homological methods in group theory 57M07 Topological methods in group theory 20F67 Hyperbolic groups and nonpositively curved groups 20E22 Extensions, wreath products, and other compositions of groups