An algebraic property of Reidemeister torsion

Abstract

For a 3‐manifold M$M$ and an acyclic SLfalse(2,double-struckCfalse)$\mathit {SL}(2,\mathbb {C})$‐representation ρ$\rho$ of its fundamental group, the SLfalse(2,double-struckCfalse)$\mathit {SL}(2,\mathbb {C})$‐Reidemeister torsion τρ(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 Efalse(Kfalse)$E(K)$, we discuss the behavior of τρ(Efalse(Kfalse))$\tau _\rho (E(K))$ when the restriction of ρ$\rho$ to the boundary torus is fixed.