Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quaternionic lattices and poly-context-free word problem

Ievgen Bondarenko

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 $Γ_τ$ 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 $Γ_τ$ 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 $\mathrm{WP}(Γ_τ)\cap a^*b^*c^*d^*$, which relies on the existence of anti-tori and certain power-type endomorphisms of the groups $Γ_τ$.

Quaternionic lattices and poly-context-free word problem

Abstract

A finitely generated group is called poly-context-free if its word problem is an intersection of finitely many context-free languages. We consider the quaternionic lattices over the field constructed by Stix-Vdovina (2017), and prove that they are not poly-context-free. As a corollary, since all the groups are quasi-isometric to , 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 , which relies on the existence of anti-tori and certain power-type endomorphisms of the groups .
Paper Structure (4 sections, 15 theorems, 27 equations)

This paper contains 4 sections, 15 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Semilinear sets and bounded context-free languages
  3. The smallest example $\Gamma_3$
  4. The groups $\Gamma_{\tau}$

Key Result

Theorem 1

The word problem in the groups $\Gamma_{\tau}$ is not poly-context-free.

Theorems & Definitions (35)

  • Theorem 1
  • Corollary 1.1
  • Proposition 2
  • proof
  • Corollary 2.1
  • proof
  • Example 1
  • Lemma 1
  • proof
  • Definition 1
  • ...and 25 more