$L^2$-torsion invariants and the Magnus representation of the mapping class group

TL;DR

This paper investigates $L^2$-torsion invariants related to the mapping class group of surfaces, providing vanishing theorems, explicit calculations, and comparisons with hyperbolic volumes.

Contribution

It introduces new $L^2$-torsion invariants for the mapping class group and computes their values, linking them to hyperbolic geometry.

Findings

Established vanishing theorems for the invariants

Explicitly calculated the first two invariants

Compared invariants with hyperbolic volumes

Abstract

In this paper, we study a series of $L^2$-torsion invariants from the viewpoint of the mapping class group of a surface. We establish some vanishing theorems for them. Moreover we explicitly calculate the first two invariants and compare them with hyperbolic volumes.