Integral approximation of simplicial volume of graph manifolds

TL;DR

This paper demonstrates that the simplicial volume of graph manifolds, which is zero, can be approximated by integral simplicial volumes of their finite coverings, providing new insights into their topological invariants.

Contribution

It introduces a method to approximate the zero simplicial volume of graph manifolds using integral volumes of finite covers, offering a uniform proof of related invariants' vanishing.

Findings

Simplicial volume of graph manifolds is zero.

Integral simplicial volumes of finite covers approximate the simplicial volume.

Vanishing of rank, Betti number, and torsion homology gradients for graph manifolds.

Abstract

Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of rank gradients, Betti number gradients and torsion homology gradients for graph manifolds.