Twisting $L^2$-invariants with finite-dimensional representations

TL;DR

This paper explores how twisting L^2-invariants like Betti numbers and torsion with finite-dimensional representations affects their properties, introducing a phi-twisted L^2-torsion function for certain CW-complexes.

Contribution

It introduces a new phi-twisted L^2-torsion function for residually finite groups with L^2-acyclic universal covers, extending the theory of L^2-invariants.

Findings

Defined phi-twisted L^2-torsion function for specific CW-complexes

Established conditions under which the function exists

Provided insights into the behavior of L^2-invariants under twisting

Abstract

We investigate how one can twist L^2-invariants such as L^2-Betti numbers and L^2-torsion with finite-dimensional representations. As a special case we assign to the universal covering of a finite connected CW-complex X together with an element phi in H^1(X;R) a phi-twisted L^2-torsion function from R^{>0} to R, provided that the fundamental group of X is residually finite and its universal covering is L^2-acyclic.