# 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.

QUESTION

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.