Groups whose word problems are not semilinear

TL;DR

This paper demonstrates that for certain classes of groups, the language of words representing the identity is not multiple context free, revealing limitations in formal language classifications of group word problems.

Contribution

It establishes non-multiple context freeness of the identity word problem language for specific groups including nilpotent, hyperbolic three-manifold, and certain right-angled Artin groups.

Findings

W is not multiple context free for nilpotent groups

W is not multiple context free for hyperbolic three-manifold groups

W is not multiple context free for certain right-angled Artin groups

Abstract

Suppose that G is a finitely generated group and W is the formal language of words defining the identity in G. We prove that if G is a nilpotent group, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then W is not a multiple context free language.