The word problem of ℤ n is a multiple context-free language

Abstract The word problem of a group G = 〈 Σ 〉 {G=\langle\Sigma\rangle} can be defined as the set of formal words in Σ * {\Sigma^{*}} 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 ℤ 2 {\mathbb{Z}^{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 ℤ n {\mathbb{Z}^{n}} is a multiple context-free language for any n.