Two-letter words and a fundamental homomorphism ruling geometric contextuality

TL;DR

This paper introduces a fundamental homomorphism linking words in a free group to symmetries in geometric structures, providing a new perspective on quantum contextuality.

Contribution

It presents a novel homomorphism that connects algebraic properties of free groups to geometric contextuality, offering a clearer understanding of the underlying structure.

Findings

Homomorphism f maps words to symmetry permutations

Reveals the core of geometric contextuality

Provides a new algebraic framework for quantum contextuality

Abstract

It has recently been recognized by the author that the quantum contextuality paradigm may be formulated in terms of the properties of some subgroups of the two-letter free group $G$ and their corresponding point-line incidence geometry $\mathcal{G}$. I introduce a fundamental homomorphism $f$ mapping the (infinitely many) words of G to the permutations ruling the symmetries of $\mathcal{G}$. The substructure of $f$ is revealing the essence of geometric contextuality in a straightforward way.