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Consider an orientable surface $S$ and its Teichmüller space $S$, which is the space of representations of its fundamental group $T(S)=\{\rho: \pi_1(S) \to \operatorname{SL}(2,\mathbb{R})\}$. Fock and Goncharov (or maybe W. Thurston?) constructed positive coordinates over this space as follows. Fix a triangulation of the surface and associate positive numbers $x$ to each edge. To each edge and corner of the triangulation we also associate matrices $B(x)$ and $I$ — whose precise form I will not write here — then any element $\gamma \in \pi_1(S)$ is represented by a word in these matrices constructed by following how the loop $\gamma$ crosses edges and corners of the triangulation.

I believe that there should be a parametrization of Teichmüller space which lies somewhat in between Fock–Goncharov–Thurston and Fenchel–Nielsen (the latter uses pants decompositions of surfaces). More precisely I think it should be possible to consider triangulations with edges that are allowed to spiral around closed loops inside (i.e. not homologous to a boundary component of) the surface. By pinching these loops, we get something that looks like a collection of triangulated surfaces attached by punctures. I imagine that one should be able to generalize the very explicit Fock–Goncharov–Thurston construction by adding matrices that describe what happens when we cross the closed loops, but I wasn't able to find by myself these matrices.

Is this something known? Is there any reference where I can learn more about this?

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  • $\begingroup$ As someone who knows nothing about the history of the subject, if it's worth calling it Fock–Goncharov–Thurston inside the body, it might be worth doing it in the title, too. It never hurts to give more credit (although I'll usually keep calling it the oscillator representation rather than the Segal–Shale–Weil representation). $\endgroup$
    – LSpice
    Commented May 7, 2022 at 15:39

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This is done, for shearing coordinates (both for pants decompositions and triangulations), by Theorem 1.3 of the paper "Shearing coordinates and convexity of length functions on Teichmüller space" by Bestvina, Bromberg, Fujiwara, and Souto. I believe that it is not very difficult to transform shearing coordinates (for a pants decomposition) into Fenchel–Nielsen coordinates. I am less sure about transforming them (for a triangulation) into Fock-Goncharov coordinates, but it should be possible.

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    $\begingroup$ Thanks, this is an interesting reference. It seems to me that in the case of a triangulation the shearing coordinates are literally the same as Fock-Goncharov-Thurston coordinates. But I don't see in this paper an explicit construction of a representation of pi_1(S) in terms of SL(2,R) matrices $\endgroup$
    – giulio bullsaver
    Commented May 7, 2022 at 17:17
  • $\begingroup$ Yes, I think you have to do that part yourself. You might have some luck asking Dylan Thurston about this - he has a preprint "On geometric intersection of curves in surfaces" which lays out how to move between the two coordinate systems on PML (the boundary of Teichmuller space). Perhaps he has thought about your exact question. $\endgroup$
    – Sam Nead
    Commented May 8, 2022 at 7:46

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