The geometry of cosets in the subgroups $$H$$H of the two-generator free group $$G=\langle a,b\rangle $$G=?a,b? nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. In previous work, it was established that dessins stabilize point-line geometries whose incidence structure reflects the commutation of (generalized) Pauli operators. Now we find that the nonexistence of a dessin for which the commutator $$(a,b)=a^{-1}b^{-1}ab$$(a,b)=a-1b-1ab precisely corresponds to the commutator of quantum observables $$[\mathcal {A},\mathcal {B}] = \mathcal {A}\mathcal {B}-\mathcal {B}\mathcal {A}$$[A,B]=AB-BA on all lines of the geometry is a signature of quantum contextuality. This occurs first at index $$|G$$|G:$$H|=9$$H|=9 in Mermin's square and at index $$10$$10 in Mermin's pentagram, as expected. Commuting sets of $$n$$n-qubit observables with $$n>3$$n>3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.
['Michel Planat']
Geometry of contextuality from Grothendieck's coset space