| Automatic Structures on Central Extensions (2007) | |||||||||||||||
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| . 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 Neumann and Reeves. 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. 1. Introduction Some years ago, we embarked on a program of computing SA(G), the set of automatic structures on a given group G up to a natural equivalence relation (definitions are given below). We had initial successes with abelian groups, geometrically finite hyperbolic groups, and graphs of groups where the edge groups are all finite. Our method in the latter two cases is to find a family C = fC i g of controlling subgroups of G. That is, each automatic structure... | |||||||||||||||
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