Geometric contextuality from the Maclachlan-Martin Kleinian groups

There are contextual sets of multiple qubits whose commutation is parametrized thanks to the coset geometry $\mathcal{G}$ of a subgroup $H$ of the two-generator free group $G=⟨x,y⟩$. One defines geometric contextuality from the discrepancy between the commutativity of cosets on $\mathcal{G}$ and that of quantum observables.It is shown in this paper that Kleinian subgroups $K=⟨f,g⟩$ that are non-compact, arithmetic, and generated by two elliptic isometries $f$ and $g$ (the Martin-Maclachlan classification), are appropriate contextuality filters. Standard contextual geometries such as some thin generalized polygons (starting with Mermin’s $3\times 3$ grid) belong to this frame. The Bianchi groups $PSL(2,O\mathrm{\_}d)$, $d\in \{1,3\}$ defined over the imaginary quadratic field $O\mathrm{\_}d=\mathbb{Q}(\sqrt{-d})$ play a special role.