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

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.