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

TL;DR

This paper studies the full language of $(\lambda,\epsilon)$-quasigeodesics in Cayley graphs and its (non-)context-freeness. It introduces the chaotic group property to obtain robust non-$\mathbb{Q}$CF results that are preserved under undistorted embeddings, and proves chaos for $\mathbb{Z}^2$, all non-virtually-cyclic finitely generated abelian groups, virtually nilpotent groups, and Baumslag--Solitar groups. The key technical tools are Ogden's lemma, subgroup distortion results (Osin), and normal-form analyses (Burillo–Elder) to construct pumping arguments via carefully crafted quasigeodesic word families. Consequently, any finitely generated group containing an undistorted copy of a non-virtually-cyclic nilpotent or a BS group is chaotic and not $\mathbb{Q}$CF, extending non-regularity from Hughes–Nairne–Spriano to a broad class of overgroups. The paper also outlines limitations and conjectures a universal pumping threshold for 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.

Figures (3)

• Figure 1: The word $w$, with pumping along dashed lines
• Figure 2: The word $t^2u_2 = t^2 a^{x_0} t^{-1} a^{x_1} t^{-1}$ traced in the Cayley graph of $BS(m,n)$. $x_1$ is, by definition, the least integer such that $y_2 = z_1+x_1$ is a multiple of $m$.
• Figure 3: The word $w_q$ traced on a projection of the Cayley graph of $BS(m,n)$. It appears to self-intersect only because of the 2D projection; the subpaths in question always lie in different sheets of the Cayley graph.

Theorems & Definitions (63)

• Definition 1.1
• Theorem 1.2: Hughes_2025HoltRees2003
• Conjecture 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 2.1: Ogden's lemma
• Corollary 2.2
• proof
• Definition 3.1: Chaotic group