Appears in collection : 2025 - T2 - WS1 - Higher rank geometric structures, Higgs bundles and physics

We classify finite orbits of the mapping class group action of the punctured sphere in $\mathrm{SL}(2,\mathbb{C})$, relying on the case of the 4-punctured sphere done by Lysovyy and Tykhyy, and on a careful understanding of compact relative character varieties of the punctured sphere in $\mathrm{SL}(2,\mathbb{R})$. This is joint work with Arnaud Maret.