Publications

Here is a list of my published research papers:

1. Action-angle coordinates for surface group representations in genus zero, J. Symplectic Geom. (2024). [arXiv:2110.13896]

   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.

2. Ergodicity of the mapping class group action on Deroin-Tholozan representations, Groups Geom. Dyn. (2022). [arXiv:2012.05775]

   Abstract: This note investigates the dynamics of the mapping class group action on the character variety of super-maximal representations of the fundamental group of a punctured sphere into $\mathrm{PSL}(2,\mathbb{R})$, discovered by Deroin and Tholozan. We apply symplectic methods developed by Goldman and Xia to prove that the action is ergodic.

Here is a list of my preprints:

1. Totally elliptic surface group representations (2024). [arXiv:2411.19748]

   Abstract: A surface group representation into a Lie group is called totally elliptic if every simple closed curve on the surface is mapped to an elliptic element of the target group. In this note, we characterize all totally elliptic surface group representations into \mathrm{PSL}_2\mathbb{R} and \mathrm{PSL}_2\mathbb{C} by showing that they are either representations into a compact subgroup or Deroin–Tholozan representations.

2. Mapping class group orbit closures for Deroin-Tholozan representations, with Y. Bouilly, G. Faraco (2024). [arXiv:2411.10269]

   Abstract: We prove that infinite mapping class group orbits are dense in the character variety of Deroin–Tholozan representations. In other words, the action is minimal except for finite orbits. Our arguments rely on the symplectic structure of the character variety, emphasizing this geometric perspective over its algebraic properties.

3. Tykhyy’s Conjecture on finite mapping class group orbits, with S. Bronstein (2024). [arXiv:2409.04379] [talk]

   Abstract: We classify the finite orbits of the mapping class group action on the character variety of Deroin–Tholozan representations of punctured spheres. In particular, we prove that the action has no finite orbits if the underlying sphere has 7 punctures or more. When the sphere has six punctures, we show that there is a unique 1-parameter family of finite orbits. Our methods also recover Tykhyy’s classification of finite orbits for 5-punctured spheres. The proof is inductive and uses Lisovyy–Tykhyy’s classification of finite mapping class group orbits for 4-punctured spheres as the base case for the induction. Our results on Deroin–Tholozan representations cover the last missing cases to complete the proof of Tykhyy’s Conjecture on finite mapping class group orbits for $\mathrm{SL}(2,\mathbb{C})$ representations of punctured spheres, after the recent work by Lam–Landesman–Litt.

4. The geometry of Deroin-Tholozan representations, with A. Fenyes (2023). [arXiv:2312.09199]

   Abstract: We present a way to build hyperbolic spheres with conical singularities by gluing together simple building blocks. Our construction provides good control over the holonomy of the resulting hyperbolic cone sphere. In particular, it can be used to realize any Deroin–Tholozan (DT) representation as the holonomy of a hyperbolic cone sphere. Our construction is inspired by the correspondence between DT representations and chains of triangles in the hyperbolic plane. It gives a geometric interpretation of certain action-angle coordinates on the space of DT representations, which come from this correspondence.

Here is a list of notes I edited:

• Proceedings of the Young Researchers Workshop on Positivity in Lie Groups, with X. Flamm (2022). [arXiv:2212.04339]

  Abstract: These notes transcribe a workshop about the notion of total positivity and $\Theta$-positivity and its relation to Higher Teichmüller Theory. $\Theta$-positivity is a notion of positivity in semisimple Lie groups and was recently introduced by Guichard and Wienhard as a generalization of Lusztig’s total positivity. It is believed to be the cathartic notion to classify higher Teichmüller spaces. Without doubt, substantial progress will be achieved in the near future on the study of $\Theta$-positive structures. These notes provide an account of the state of the art as of 2021.

• A note on character varieties, available here (will be regularly updated).

  Abstract: These notes are an extended version of a series of mini-courses that I gave in the spring of 2022 at the KIT Karlsruhe and at the University of Heidelberg and in the spring of 2024 at Seoul National University. The main objective is to introduce the notions of representation varieties and character varieties from the perspective of both differential geometry and algebraic geometry. It is common to encounter different definitions of character varieties in the literature, depending on each author’s “favourite quotient” which varies depending on the context and the applications.

Here is a link to my PhD thesis:

• The symplectic geometry of surface group representations in genus zero, 2022. Available here.

Here is a link to my master thesis:

• Forcing relations for periodic orbits of surface homeomorphisms, 2018. Available here.