Noncommutative marked surfaces

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