hyperbolic geometry – Cluster algebras of type A and X

I will base my question on Fock and Goncharov’s paper (https://arxiv.org/pdf/math/0510312).

Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without marked points. We collectively call punctures and unmarked boundaries holes.

To $S$ we can associate two Teichmuller spaces, let me call them $T_A(S)$ and $T_X(S)$.
The first is the Decorated Teichmuller space, where above each puncture and marked point is made a choice of an horocycle. In this case all holes must be punctures.
The second is the usual Teichmuller space, and we now allow un-marked boundaries.

We define a triangulation $T$ of $S$ to be a triangulation of the surface obtained by replacing each hole with a puncture.

Associated to a triangulation $T$ of $S$ there are coordinate systems for both $T_A(S)$ and $T_X(S)$. They are positive numbers associated to the edges $e$ of $T$ and to the non-boundary edges of $T$ respectively. They are given by $lambda$-lengths for $T_A(S)$ (indicated by $lambda_e$) and by shear coordinates for $T_X(S)$ (indicated by $x_e$).

There is a map $T_A(S) to T_X(S)$, which I will call the shear map, which is given in coordinates by $x_e = frac{lambda_a lambda_c}{lambda_b lambda_d}$, for each inner edge $e$ belonging to triangles $(e,a,b)$ and $(e,c,d)$ of $T$.

This map lands in the subspace $prod_{e in p} x_e = 1$, where the product is taken over all edges $e$ incident to a hole $p$. This is a subspace of $T_X(S)$ because in general this product gives the exponential of the hyperbolic length of the hole.


Is it possible to define a “natural” map $T_A(S) to T_X(S)$ that doesn’t land in this subspace?
By “natural” I mean that it does not depend on any choice of triangulation.