Anti-tori in quaternionic lattices over \(\mathbb{F}_q(t)\)

Abstract

An anti-torus in a CAT(0) group is a subgroup \(\langle a,b\rangle\), where \(a\) and \(b\) do not have commuting powers. We study anti-tori in quaternionic lattices \(\Gamma_\tau\) over the field \(\mathbb{F}_q(t)\) introduced by Stix-Vdovina (2017). We determine when every pair of generators of \(\Gamma_\tau\) generates an anti-torus, and establish the existence of \(a,b\in\Gamma_\tau\) such that the subgroup \(\langle a^{p^n},b^{p^n}\rangle\) is not abelian and not free for all \(n\geq 0\). Explicit examples of matrices \(a,b\in SL_3(\mathbb{F}_q(t))\) with this property are given.