arXiv subjects
Publications and source records attributed to Thomas Le Fils.
We characterise the elements of $H^1(S, Z, \mathbb C)$, where $S$ is a closed surface and $Z\subset S$ is a finite set, that arise as the relative periods of an abelian differential in a given connected component of a stratum of their moduli space. This generalises a theorem obtained independently by Bainbridge, Johnson, Judge and Park and the author, and answers a question of Filip.
We classify the Teichmüller curves in the moduli space of genus three Riemann surfaces $\mathcal M_3$ that are obtained by a covering construction from a primitive Teichmüller curve in $\mathcal M_2$. We describe the action on homology modulo two of the affine groups of translation surfaces generating these primitive curves. We also classify the $\mathrm{SL}_2(\mathbb Z)$-orbits of square-tiled surfaces in $\mathcal H(2, 2)$ that are a cover of a genus two one.
For any irreducible representation of a surface group into $\mathrm{SL}_2(\mathbb{C})$, we show that there exists a pants decomposition where the restriction to any pair of pants is irreducible and where no curve of the decomposition is sent to a trace $\pm 2$ element. We prove a similar property for $\mathrm{SO}_3$-representations. We also investigate the type of pants decomposition that can occur in this setting for a given representation. This result was announced in a previous paper of the first and third named authors, motivated by the study of the Azumaya locus of the skein algebra of surfaces at roots of unity.
We characterize the representations of the fundamental group of a closed surface to $\mathrm{PSL}_2(\mathbb C)$ that arise as the holonomy of a branched complex projective structure with fixed branch divisor. In particular, we compute the holonomies of the spherical metrics with prescribed integral conical angles and the holonomies of affine structures with fixed conical angles on closed surfaces.
We give a necessary and sufficient condition for a representation of the fundamental group of a closed surface of genus at least 2 to $\mathbb C$ to be the holonomy of a translation surface with a prescribed list of conical singularities. Equivalently, we determine the period maps of abelian differentials with prescribed list of multiplicites of zeros. Our main result was also obtained, independently, by Bainbridge, Johnson, Judge and Park.
We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb C)$-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to $\mathrm{PSL}_2 (\mathbb C)$ is the holonomy of a projective structure, possibly with one branched point of order $2$. We show that pentagon representations arise as such holonomies and repair their proof.