# Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces.

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 Teichmüller spaces, let me call them $$T_A(S)$$ and $$T_X(S)$$. The first is the decorated Teichmüller space, where above each puncture and marked point is made a choice of a horocycle. In this case all holes must be punctures. The second is the usual Teichmüller 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.

One generalizes $$\lambda$$-lengths by adding to each an index that represents on which level of the spiral we are.