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

TL;DR

This paper proves that for any irreducible surface group representation into SL2(C), there exists a pants decomposition with irreducible restrictions and no curves mapped to trace ±2, aiding the understanding of surface representations.

Contribution

It establishes the existence of compatible pants decompositions with specific irreducibility and trace conditions for SL2(C) and SO3 representations, advancing the study of surface group representations.

Findings

Existence of pants decompositions with irreducible restrictions

No curves mapped to trace ±2 in the decomposition

Applicable to both SL2(C) and SO3 representations

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.