  Using Fock--Goncharov higher Teichm\"uller space variables we derive Darboux
coordinate representation for entries of general symplectic leaves of the
$\mathcal A_n$ groupoid of upper-triangular matrices and, in a more general
setting, of higher-dimensional symplectic leaves for algebras governed by the
reflection equation with the trigonometric $R$-matrix. The obtained results are
in a perfect agreement with the previously obtained Poisson and quantum
representations of groupoid variables for $\mathcal A_3$ and $\mathcal A_4$ in
terms of geodesic functions for Riemann surfaces with holes. We represent
braid-group transformations for $\mathcal A_n$ via sequences of cluster
mutations in the special $\mathbb A_n$-quiver. We prove the groupoid relations
for quantum transport matrices and, as a byproduct, obtain the Goldman bracket
in the semiclassical limit.
