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# Abelianization of SL(2,R) local systems

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## Abstract

The "abelianization" process introduced by Gaiotto, Hollands, Moore, and Neitzke turns $\operatorname{SL}_K \mathbb{C}$ local systems on a punctured surface into $\mathbb{C}^\times$ local systems, giving coordinates on the decorated $\operatorname{SL}_K \mathbb{C}$ character variety that are known to match Fock and Goncharov's cluster coordinates in the $\operatorname{SL}_2 \mathbb{C}$ case. This paper extends abelianization to $\operatorname{SL}_2 \mathbb{R}$ local systems on compact surfaces, using tools from dynamics to overcome the technical challenges that arise in the compact case. The approach taken here seems to complement the one recently used by Bonahon and Dreyer to arrive at a similar construction in a different geometric setting.

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... This case is more difficult, since problems of dynamics mix with the combinatorial problems. Some of the necessary tools in case K = 2 have been developed in [14]. ...
... Again see Figure 7 and Figure 11 for illustrative examples. 14 These marked points divide the circle at infinity into 2n + 6 arcs. Call an arc at infinity initial (final) if its two boundary points have the same initial (final) label; there are n + 3 arcs of each type. ...
... Define an abelianization tree compatible with W (P 0 , ϑ) to be a collection of oriented arcs in C, with each arc labeled by a sheet of Σ and a representation of SL(3, R) (either fundamental V or its dual V * ), with the following properties: 14 In comparing the asymptotics in those figures to the description above, one must keep in mind the permutations of sheets which occur when one crosses the branch cuts. For example, in Figure 7, starting from the rightmost marked point just above the branch cut, we see the sequence 31, 21, 23, 13, 12; the next label would ordinarily be 32, but we also cross a branch cut which induces the permutation (123), so the next label is instead 13. 15 Strictly speaking, what [2,3,4] predict is a local coordinate system on a patch of a complexification of • Each arc has two endpoints. ...
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The dynamics of billiard flows in rational polygons of dynamical systems
• J Smillie
J. Smillie, "The dynamics of billiard flows in rational polygons of dynamical systems," in Dynamical Systems, Ergodic Theory and Applications, Y. Sinai, ed., ch. 11. 2000.
MH 7750: Set theoretic topology
• G Gruenhage
G. Gruenhage, " MH 7750: Set theoretic topology. " http://www.auburn.edu/~gruengf/fall14.html, 2014.
Potentially cluster-like coordinates from dense spectral networks https://www.ma.utexas.edu/users/afenyes/writing.html. Poster presented at Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes
• A Fenyes
A. Fenyes, " Potentially cluster-like coordinates from dense spectral networks. " https://www.ma.utexas.edu/users/afenyes/writing.html. Poster presented at Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes. July 2015, CRM.
Moduli spaces of local systems and higher Teichmüller theory
• V Fock
• A Goncharov
V. Fock and A. Goncharov, "Moduli spaces of local systems and higher Teichmüller theory," arXiv:math/0311149 [math.AG].
• L Hollands
• A Neitzke
L. Hollands and A. Neitzke, "Spectral networks and Fenchel-Nielsen coordinates," arXiv:1312.2979 [math.GT].
Hitchin characters and geodesic laminations
• F Bonahon
• G Dreyer
F. Bonahon and G. Dreyer, "Hitchin characters and geodesic laminations," arXiv:1410.0729v2 [math.GT].
of Cambridge studies in advanced mathematics
• M Viana
M. Viana, Lectures on Lyapunov Exponents, vol. 145 of Cambridge studies in advanced mathematics. Cambridge University Press, 2014.
Poster presented at Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes
• A Fenyes
A. Fenyes, "Potentially cluster-like coordinates from dense spectral networks." https://www.ma.utexas.edu/users/afenyes/writing.html. Poster presented at Positive Grassmannians: Applications to integrable systems and super Yang-Mills scattering amplitudes. July 2015, CRM.