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...

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Bibliographic Details
Published in:arXiv.org 2022-07
Main Authors: Kitano, Teruaki, Nozaki, Yuta
Format: Article
Language:English
Subjects:
Algebra
Manifolds
Representations
Toruses
Online Access:Get full text
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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