Quasigeodesic languages are not context-free in some non-hyperbolic groups

TL;DR

This paper proves that in certain non-hyperbolic groups, the language of quasigeodesics with fixed error constants is not context-free, extending previous results about regularity failure.

Contribution

It demonstrates that for specific non-hyperbolic groups, the quasigeodesic language cannot be context-free, broadening understanding of language complexity in geometric group theory.

Findings

Quasigeodesic languages are not context-free in non-virtually-cyclic nilpotent groups.

The result extends to groups containing such groups as undistorted subgroups.

The conclusion strengthens previous results on regularity failure in non-hyperbolic groups.

Abstract

We study the full language of quasigeodesics in Cayley graphs, with fixed error constants. We show that, given a non-virtually-cyclic nilpotent group or Baumslag--Solitar group, and any finite generating set, such languages fail to be context-free for sufficiently large error constants. In fact, this conclusion holds for any finitely generated group which contains one of these groups as an undistorted subgroup. This strengthens a recent theorem of Hughes, Nairne, and Spriano, who showed that such languages fail to be regular in any non-hyperbolic group, for sufficiently large error constants.