Noncommutative marked surfaces

The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface $\Sigma$. This is a noncommutative algebra ${\mathcal A}_\Sigma$ generated by "noncommutative geodesics" between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy-Pl\"ucker relations. It turns out that the algebra ${\mathcal A}_\Sigma$ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of $\Sigma$, which confirms its "cluster nature". As a surprising byproduct, we obtain a new topological invariant of $\Sigma$, which is a free or a 1-relator group easily computable in terms of any triangulation of $\Sigma$. Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.