  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.
