The asymptotic behavior of the Reidemeister torsion for Seifert manifolds and PSL(2;R)-representations of Fuchsian groups

TL;DR

This paper investigates how PSL(2;R)-representations of Fuchsian groups influence the asymptotic behavior of Reidemeister torsion in Seifert manifolds, revealing limits tied to the Euler class and characteristic of associated orbifolds.

Contribution

It establishes a connection between PSL(2;R)-representations, Euler classes, and the asymptotics of Reidemeister torsion in Seifert manifolds, including explicit limit formulas.

Findings

Limit of leading coefficient determined by Euler class.

Reidemeister torsion for unit tangent bundle converges to -χ log 2.

Relation between Z_2-extensions and torsion asymptotics.

Abstract

We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PSL(2;R)-representation of a Fuchsian group. In particular, the leading coefficient of the Reidemeister torsion for the unit tangent bundle over a two-orbifold converges to $-\chi \log2$ where $\chi$ is the Euler characteristic of the two-orbifold. We also give a relation between $\mathbb{Z}_2$-extensions for PSL(2;R)-representations of a Fuchsian group and the asymptotics of the Reidemeister torsion.