Approximating L^2-invariants and homology growth

TL;DR

This paper investigates the asymptotic behavior of various homological invariants, showing they vanish under certain fibering conditions or group properties, with implications for both topological spaces and groups.

Contribution

It establishes new results on the vanishing of L^2-invariants and homology growth for spaces and groups with specific fibering or group-theoretic structures.

Findings

Homology invariants vanish in the limit for fibered CW-complexes.

Aspherical manifolds with S^1-actions exhibit vanishing invariants.

Groups with infinite normal elementary amenable subgroups show similar vanishing behavior.

Abstract

In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S^1-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups.