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Action-angle coordinates for surface group representations in genus zero

Arnaud Maret

Abstract

We study a compact family of totally elliptic representations of the fundamental group of a punctured sphere into $\mathrm{PSL}(2,\mathbb R)$ discovered by Deroin and Tholozan and named after them. We describe a polygonal model that parametrizes the relative character variety of Deroin--Tholozan representations in terms of chains of triangles in the hyperbolic plane. We extract action-angle coordinates from our polygonal model as geometric quantities associated to chains of triangles. The coordinates give an explicit isomorphism between the space of representations and the complex projective space. We prove that they are almost global Darboux coordinates for the Goldman symplectic form.

Action-angle coordinates for surface group representations in genus zero

Abstract

We study a compact family of totally elliptic representations of the fundamental group of a punctured sphere into discovered by Deroin and Tholozan and named after them. We describe a polygonal model that parametrizes the relative character variety of Deroin--Tholozan representations in terms of chains of triangles in the hyperbolic plane. We extract action-angle coordinates from our polygonal model as geometric quantities associated to chains of triangles. The coordinates give an explicit isomorphism between the space of representations and the complex projective space. We prove that they are almost global Darboux coordinates for the Goldman symplectic form.

Paper Structure

This paper contains 23 sections, 39 theorems, 161 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The results
  3. Organization of the paper
  4. Acknowledgements
  5. Preliminaries
  6. Volume of a representation
  7. Remarkable connected components
  8. A polygonal model
  9. The case of the thrice-punctured sphere
  10. The general case
  11. The torus action revisited
  12. Complex projective coordinates
  13. Definition of the map
  14. A Wolpert-type formula
  15. Proof of Theorem \ref{['thm:symplectomorphism']}
  16. ...and 8 more sections

Key Result

Theorem A

If we set $\sigma_i\coloneqq\gamma_1+\ldots+\gamma_i$, then are action-angle coordinates for each Deroin-Tholozan relative character variety.

Figures (12)

  • Figure 1: On top: a pants decomposition of a sphere with six punctures into four pairs of pants. This illustration is modelled on DeTh19. On bottom: a corresponding chain of geodesic triangles in the upper half-plane. The angles between consecutive triangles in the chain are denoted by $\gamma_i$.
  • Figure 2: The various aspects of Deroin-Tholozan representations are drawn as a diagram. The space $\overline{\textnormal{Hyp}}_{\alpha}$ refers to the space of hyperbolic metrics with conical singularities introduced at the end of Section \ref{['sec:results']}. It projects to $\mathop{\mathrm{Rep}}\nolimits^{\textnormal{\tiny{DT}}}_\alpha(\Sigma_n,G)\xspace$ as mentioned earlier. The projection corresponds to the outer dashed arrows.
  • Figure 3: The case of a 6-punctured sphere: The simple closed curves $b_1,b_2,b_3$ and the peripheral curves $c_1,\ldots,c_6$.
  • Figure 4: Example of a configuration of the fixed points and the associated chain of triangles in the case $n=6$.
  • Figure 5: The two non-degenerate configurations of fixed points. On the left: the configuration where $\Delta(C_1,C_2,C_3)$ is clockwise oriented and the interior angles are $\pi-\alpha_i/2$. On the right: the configuration where $\Delta(C_1,C_2,C_3)$ is anti-clockwise oriented and the interior angles are $\alpha_i/2$.
  • ...and 7 more figures

Theorems & Definitions (69)

  • Theorem A
  • Theorem B: see Theorem \ref{['thm:symplectomorphism']}
  • Theorem C: see Corollary \ref{['cor:analogue-Wolpert-formula']}
  • Definition 2.2
  • Theorem 2.3: BIW10
  • Definition 2.4
  • Definition 2.6
  • Theorem 2.7: DeTh19
  • Remark 2.8
  • Proposition 2.9: total ellipticity
  • ...and 59 more