Quaternionic lattices and the poly-context-free word problem

Abstract A finitely generated group 𝐺 is said to be poly-context-free if its word problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>WP</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathrm{WP}(G) is an intersection of finitely many context-free languages. We consider the quaternionic lattices <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>τ</m:mi> </m:msub> </m:math> \Gamma_{\tau} over the field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>τ</m:mi> </m:msub> </m:math> \Gamma_{\tau} are quasi-isometric to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo lspace="0.222em" rspace="0.222em">×</m:mo> <m:msub> <m:mi>F</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>WP</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>τ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>∩</m:mo> <m:mrow> <m:msup> <m:mi>a</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mi>b</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mi>c</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mi>d</m:mi> <m:mo>∗</m:mo> </m:msup> </m:mrow> </m:mrow> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>τ</m:mi> </m:msub> </m:math> \Gamma_{\tau} .