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The \emph{word problem} of a group $G = \langle \Sigma \rangle$ can bedefined as the set of formal words in $\Sigma^*$ that represent the identity in$G$. When viewed as formal languages, this gives a strong connection betweenclasses of groups and classes of formal languages. For example, Anisimov showedthat 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 itsword problem is a context-free language. Above this, not much was known, untilSalvati showed recently that the word problem of $\mathbb{Z}^2$ is a multiplecontext-free language, giving first such example. We generalize Salvati'sresult to show that the word problem of $\mathbb{Z}^n$ is a multiplecontext-free language for any $n$.

The Word Problem of $\mathbb{Z}^n$ Is a Multiple Context-Free Language