$L^2$-torsion of automorphisms

TL;DR

This paper develops the theory of $L^2$-torsion for automorphisms of groups, providing formulas and computations for various classes of groups including hyperbolic, CAT(0) lattices, and graph manifolds.

Contribution

It introduces a new framework for $L^2$-torsion of automorphisms and derives a combination formula, enabling explicit calculations for complex group structures.

Findings

Computed $L^2$-torsion for hyperbolic automorphisms

Derived a combination formula for $L^2$-torsion in group actions

Calculated $L^2$-torsion for CAT(0) lattices and graph manifolds

Abstract

We develop the theory of $L^2$-torsion of an automorphism of a group and compute it for every automorphism of a group which is hyperbolic and one-ended relative to a finite collection of virtually polycyclic groups. We also prove a combination formula for the $L^2$-torsion of a group in terms of the $L^2$-torsion of its stabilisers of a sufficiently nice action on a contractible space. We apply it to compute the $L^2$-torsion of a selection of CAT(0) lattices, of many relatively hyperbolic groups and their automorphisms, of higher dimensional graph manifolds, and of handlebody groups.