Quaternionic lattices and poly-context-free word problem

TL;DR

This paper proves that certain quaternionic lattice groups over function fields do not have poly-context-free word problems, revealing that this property is not preserved under quasi-isometries, and advances understanding of group language complexity.

Contribution

It demonstrates that quaternionic lattices over function fields are not poly-context-free, showing the class is not closed under quasi-isometries, and introduces new techniques involving anti-tori and endomorphisms.

Findings

Quaternionic lattices are not poly-context-free.

Poly-context-freeness is not preserved under quasi-isometries.

Language analysis relies on anti-tori and endomorphisms.

Abstract

A finitely generated group $G$ is called poly-context-free if its word problem $\mathrm{WP}(G)$ is an intersection of finitely many context-free languages. We consider the quaternionic lattices $\Gamma_\tau$ over the field $\mathbb{F}_{q}(t)$ constructed by Stix-Vdovina (2017), and prove that they are not poly-context-free. As a corollary, since all the groups $\Gamma_{\tau}$ are quasi-isometric to $F_2\times F_2$, the class of groups with poly-context-free word problem is not closed under quasi-isometries. The result follows from the description of the language $\mathrm{WP}(\Gamma_\tau)\cap a^*b^*c^*d^*$, which relies on the existence of anti-tori and certain power-type endomorphisms of the groups $\Gamma_\tau$.