Quaternionic lattices and poly-context-free word problem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916122148732928 |
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| author | Bondarenko, Ievgen |
| author_facet | Bondarenko, Ievgen |
| contents | 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 $Γ_τ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07494 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quaternionic lattices and poly-context-free word problem Bondarenko, Ievgen Group Theory 20F10, 68Q45, 20F65, 20E08 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 $Γ_τ$. |
| title | Quaternionic lattices and poly-context-free word problem |
| topic | Group Theory 20F10, 68Q45, 20F65, 20E08 |
| url | https://arxiv.org/abs/2402.07494 |