Noncommutative marked surfaces

Abstract

The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface . This is a noncommutative algebra A 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 A exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of , which confirms its “cluster nature”. As a surprising byproduct, we obtain a new topological invariant of , which is a free or a 1-relator group easily computable in terms of any triangulation of . Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.