Twisted $L^2$-torsion on the character variety

TL;DR

This paper introduces a twisted $L^2$-torsion invariant on the character variety of 3-manifolds and proves its real analytic nature near holonomy representations for hyperbolic finite-volume cases.

Contribution

It defines a new twisted $L^2$-torsion invariant on 3-manifold character varieties and analyzes its properties, especially its analyticity near specific representations.

Findings

$L^2$-torsion is real analytic near holonomy representations

The invariant is defined on the character variety of 3-manifolds

Properties of the twisted $L^2$-torsion are studied in hyperbolic cases

Abstract

We define a twisted $L^2$-torsion on the character variety of 3-manifold $M$ and study some of its properties. In the case where $M$ is hyperbolic of finite volume, we prove that the $L^2$-torsion is a real analytic function on a neighborhood of any lift of the holonomy representation.