This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties.Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden.Berenstein, ArkadyRetakh, Vladimir2019-06-2820151864-7596https://doi.org/10.34657/2752https://oa.tib.eu/renate/handle/123456789/2773The 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.application/pdfeng510Report