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.


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.


1 Answer 1


I believe there is a natural extension of these ideas by considering an infinite spiralling horocycle at the hole.

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

The shear map is built by taking cross ratios respecting the spiral levels.

  • $\begingroup$ Have you read Fomin-Shapiro-Thurston's second paper arxiv.org/abs/1210.5569 ? I think this spiraling construction appears there. (I don't know the answer to your question, though.) $\endgroup$ Jun 9, 2021 at 13:59
  • $\begingroup$ I thought they replace the puncture with a boundary with a marked point. $\endgroup$ Jun 9, 2021 at 14:00
  • $\begingroup$ I'll admit I wrote this comment without rereading the paper, on the basis of a conversation I remember with Dylan, near when the paper come out, when he drew the infinite spiral for me. Let me see if I can find it in the paper. $\endgroup$ Jun 9, 2021 at 14:03
  • $\begingroup$ See Figure 23 (I still haven't reread the paper; just skimmed looking for the picture Dylan drew). $\endgroup$ Jun 9, 2021 at 14:04

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