Adjoint Reidemeister torsion of 3-manifolds with torus boundary for semisimple algebraic groups

TL;DR

This paper extends the concept of adjoint Reidemeister torsion to 3-manifolds with torus boundaries for semisimple algebraic groups, providing formulas and computations for specific cases like the figure-eight knot complement.

Contribution

It introduces a generalized adjoint torsion function for G-local systems on 3-manifolds, extending Porti's work for SL_2, and computes explicit torsions for PGSp_4 in knot complements.

Findings

Defined the adjoint torsion function for G-local systems on 3-manifolds.

Proved the regularity condition for hyperbolic structures via principal embeddings.

Computed explicit PGSp_4-torsions for the figure-eight knot complement.

Abstract

Let $M$ be a compact oriented $3$-manifold with boundary consisting of tori, and let $G$ be a semisimple algebraic group. We define the adjoint torsion function on the moduli stack of $G$-local systems on $M$ satisfying a certain regularity condition, extending the construction by Porti for $G = \mathrm{SL}_2$. When $M$ is a cusped hyperbolic manifold, we prove that the local system associated with the image of the complete hyperbolic structure via a principal embedding $\mathrm{PGL}_2 \to G$ satisfies the regularity condition. Moreover, we provide a formula expressing its adjoint torsion as a product of $\mathrm{PGL}_2$-torsions associated with the simple $\mathrm{PGL}_2$-modules with multiplicity given by the exponents of the Lie algebra of $G$. We compute the adjoint $\mathrm{PGSp}_4$-torsions of the figure-eight knot complement for two boundary-unipotent local systems, one is arising from the complete hyperbolic structure via a principal embedding, and the other is defined over a number field of degree $6$ and not arising from any $\mathrm{PGL}_2$-local system via principal embeddings.