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The goal of my talk (based on a recent joint work with Vladimir Retakh) is to attach a noncommutative cluster-like structure to each each marked surface Sigma. This is a noncommutative algebra generated by "noncommutative geodesics" between marked points subject to certain triangular relations and noncommutative analogues of Ptolemy-Plucker relations. It turns out that the algebra exhibit a noncommutative Laurent Phenomenon with respect to any triangulation of Sigma, which confirms it "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.