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

Tsukasa Ishibashi, Yuma Mizuno

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.

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

Abstract

Let be a compact oriented -manifold with boundary consisting of tori, and let be a semisimple algebraic group. We define the adjoint torsion function on the moduli stack of -local systems on satisfying a certain regularity condition, extending the construction by Porti for . When is a cusped hyperbolic manifold, we prove that the local system associated with the image of the complete hyperbolic structure via a principal embedding satisfies the regularity condition. Moreover, we provide a formula expressing its adjoint torsion as a product of -torsions associated with the simple -modules with multiplicity given by the exponents of the Lie algebra of . We compute the adjoint -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 and not arising from any -local system via principal embeddings.
Paper Structure (38 sections, 46 theorems, 179 equations, 5 figures, 1 table)

This paper contains 38 sections, 46 theorems, 179 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Adjoint Reidemeister torsion for $G_{\mathbb{C}} = \mathop{\mathrm{SL}}\nolimits_2(\mathbb{C})$
  4. Main results
  5. Definition of the adjoint torsion function
  6. Principal embedding of the geometric representation
  7. Adjoint torsion of a local system not arising from $\mathop{\mathrm{PGL}}\nolimits_2$-local systems via principal embeddings
  8. Related works
  9. Future work: computation of adjoint torsion by cluster coordinates
  10. Determinants of complexes
  11. Graded invertible modules
  12. Determinant functors
  13. Moduli stack of local systems and twisted complexes
  14. Moduli stack of $G$-local systems
  15. Moduli stack of linear local systems
  16. ...and 23 more sections

Key Result

Theorem 1

We define the adjoint torsion function as a morphism of stacks whose value $\mathop{\mathrm{\mathsf{tor}}}\nolimits_{G, M, \gamma, \mathfrak{o}}^{\mathrm{Ad}}(P) \in K$ at a boundary-adjoint-regular local system $(K, P) \in \lvert \mathop{\mathrm{Loc}}\nolimits_{G, M}^{\partial\text{-}\mathrm{Ad}\text{-}\mathrm{reg}} \rvert$ is defined as a non-acyclic Reidemei

Figures (5)

  • Figure 1: The map \ref{['eq:attaching_sphere']}.
  • Figure 2: An example of $2$-dimensional CW copmlex.
  • Figure 3: Ideal triangulation of the figure-eight knot complement.
  • Figure 4: The CW structure of the figure-eight knot exterior obtained by truncating vertices in the ideal triangulation in \ref{['fig:figure_eight_triangulation']}.
  • Figure 5: The CW structure of the boundary torus. In the upper figure, the endpoints of $E_0$ are shown as solid points, and the endpoints of $E_1$ are shown as circled points. The lower figure indicates where each cell lies in the tetrahedra. The label $i(j)$ on a triangle indicates that the triangle is obtained by truncating the $j$-th vertex of the $i$-th tetrahedron. If a corner of the triangle $i(j)$ is labeled by $k$, this means that the vertex corresponding to that corner is an endpoint of the edge $ik$ in the $i$-th tetrahedron.

Theorems & Definitions (106)

  • Theorem 1: Definition \ref{['def:adjoint torsion function']}
  • Theorem 2: \ref{['thm:geom_regular']}
  • Theorem 3: \ref{['thm:adjoint_torsion_hol']}
  • Example 1
  • Lemma 1
  • proof
  • Definition 1
  • Example 1
  • Theorem 1: Knudsen
  • Theorem 2: Knudsen
  • ...and 96 more