Compatible pants decompositions for $\mathrm{SL}_2(\mathbb{C})$-representations of surface groups

Renaud Detcherry; Thomas Le Fils; Ramanujan Santharoubane

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arXiv · 2210.09854

Compatible pants decompositions for $\mathrm{SL}_2(\mathbb{C})$-representations of surface groups

Renaud Detcherry·Thomas Le Fils·Ramanujan Santharoubane

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Abstract

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.

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Renaud Detcherry, Thomas Le Fils, Ramanujan Santharoubane. 2022-10-18. Compatible pants decompositions for $\mathrm{SL}_2(\mathbb{C})$-representations of surface groups. https://arxiv.org/abs/2210.09854

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