| Walter D. Neumann and Michael Shapiro (2007) | |||||||||||||
Abstract | |||||||||||||
| . We describe a virtually abelian group G generated by a finite set X such that there is no regular language of geodesic X-words that surjects to G by evaluation. 1. Introduction Cannon gave an example in [ECHLPT] (Example 4.4.2, see also [NS]) of a virtually abelian group with finite generating set X such that the language of geodesic X-words for G is not a regular language. Since his example admits a geodesic automatic structure in the generators X, it left open the possibility that this might always be so: a virtually abelian group might admit a geodesic automatic structure for any generating set, or at least a geodesic regular unique normal form (which is weaker). This is suggested as a question in [ECHLPT]. A reason to hope it might be true was that it would give a very satisfactory proof of Benson's theorem [B] that the growth function of a virtually abelian group is rational with respect to any generating set. In this note we exhibit a virtually abelian group G with finite gen... | |||||||||||||
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