An algebraic property of Reidemeister torsion

TL;DR

This paper investigates the algebraic properties of Reidemeister torsion for 3-manifolds, demonstrating that it is often an algebraic integer and exploring its behavior in specific cases like Seifert fibered spaces and hyperbolic manifolds.

Contribution

It proves that Reidemeister torsion is an algebraic integer for many 3-manifolds and analyzes its behavior for knot exteriors with fixed boundary conditions.

Findings

Reidemeister torsion is algebraic for most Seifert fibered spaces.

Reidemeister torsion is an algebraic integer for infinitely many hyperbolic 3-manifolds.

Behavior of torsion for knot exteriors with fixed boundary restrictions.

Abstract

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.