An algebraic property of Reidemeister torsion

For a 3-manifold \(M\) and an acyclic \(\mathit{SL}(2,\mathbb{C})\)-representation \(\rho\) of its fundamental group, the \(\mathit{SL}(2,\mathbb{C})\)-Reidemeister torsion \(\tau_\rho(M) \in \mathbb{C}^\times\) is defined. If there are only finitely many conjugacy classes of irreducible representat...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Authors Kitano, Teruaki, Nozaki, Yuta
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.07.2022
Subjects
Algebra
Manifolds
Mathematics - Geometric Topology
Representations
Toruses
Online AccessGet full text

Cover

More Information
Summary:For a 3-manifold \(M\) and an acyclic \(\mathit{SL}(2,\mathbb{C})\)-representation \(\rho\) of its fundamental group, the \(\mathit{SL}(2,\mathbb{C})\)-Reidemeister torsion \(\tau_\rho(M) \in \mathbb{C}^\times\) is defined. If there are only finitely many conjugacy classes of irreducible representations, then the Reidemeister torsions are known to be algebraic numbers. Furthermore, we prove that the Reidemeister torsions are not only algebraic numbers but also algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3-manifolds. Also, for a knot exterior \(E(K)\), we discuss the behavior of \(\tau_\rho(E(K))\) when the restriction of \(\rho\) to the boundary torus is fixed.
ISSN:2331-8422
DOI:10.48550/arxiv.2201.01400