An algebraic property of Reidemeister torsion. (14th September 2022)

• Record Type: Journal Article
• Authors: Kitano, Teruaki; Nozaki, Yuta
• Is Part Of: Transactions of the London Mathematical Society. Volume 9:Number 1(2022)
• Page Start: 136
• Page End: 157
• Publication Date: 2022-09-14
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1112/tlm3.12049
• Languages: English
• ISSNs: 2052-4986

Abstract

Abstract: For a 3‐manifold M $M$ and an acyclic SL ( 2, C ) $\mathit {SL}(2, \mathbb {C})$ ‐representation ρ $\rho$ of its fundamental group, the SL ( 2, C ) $\mathit {SL}(2, \mathbb {C})$ ‐Reidemeister torsion τ ρ ( M ) ∈ C × $\tau _\rho (M) \in \mathbb {C}^\times$ 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 ) $E(K)$, we discuss the behavior of τ ρ ( E ( K ) ) $\tau _\rho (E(K))$ when the restriction of ρ $\rho$ to the boundary torus is fixed.