$L^2$-torsion of automorphisms

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.

arXiv comment40 pages, final version, to appear in Journal of the LMS