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The word problem of ℤn is a multiple context-free language

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Abstract

The word problem of a group G=Σ can be defined as the set of formal words in Σ* that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of 2 is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of n is a multiple context-free language for any n.

MSC 2010: 68Q45; 20K15

Award Identifier / Grant number: DMS-1440140

Funding statement: The material is based upon work supported by the National Science Foundation under Grant no. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2016 semester.

Acknowledgements

The author would like to thank Robert Gilman for introducing him to this topic, Koji Fujiwara for pointing him to Burago’s lemma, Tullia Dymarz and Moon Duchin for useful discussions and their support, and the anonymous referees for their very useful comments and suggestions.

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Received: 2017-9-22
Published Online: 2018-4-11
Published in Print: 2018-5-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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