L^2-Invariants of Finite Aspherical CW-Complexes

TL;DR

This paper studies $L^2$-invariants of finite aspherical CW-complexes with specific fundamental group structures, proving positivity of Novikov-Shubin invariants and vanishing of $L^2$-torsion under certain conditions.

Contribution

It establishes new results on the positivity of Novikov-Shubin invariants and the vanishing of $L^2$-torsion for a class of aspherical CW-complexes with elementary amenable subgroups.

Findings

Novikov-Shubin invariants $oldsymbol{ ext{are positive}}$ for the complexes.

$L^2$-torsion $oldsymbol{ ext{vanishes}}$ if the fundamental group has semi-integral determinant.

Provides conditions under which these $L^2$-invariants behave predictably.

Abstract

Let $X$ be a finite aspherical CW-complex whose fundamental group $\pi_1(X)$ possesses a subnormal series $\pi_1(X) \rhd G_m \rhd ... \rhd G_0$ with a non-trivial elementary amenable group $G_0$. We investigate the $L^2$-invariants of the universal covering of such a CW-complex $X$. We show that the Novikov-Shubin invariants $\alpha_n({\tilde X})$ are positive. We further prove that the $L^2$-torsion $\rho^{(2)}({\tilde X})$ vanishes if $\pi_1(X)$ has semi-integral determinant.