Simplicial volume of manifolds with amenable fundamental group at infinity

TL;DR

This paper proves that open manifolds with certain amenability conditions at infinity have finite or zero simplicial volume, extending understanding of how fundamental group properties influence manifold invariants.

Contribution

It establishes the finiteness and vanishing of simplicial volume for manifolds with amenable fundamental groups at infinity, including cases with multiple ends and simple connectivity at infinity.

Findings

Finite simplicial volume for manifolds with amenable fundamental group at infinity.

Vanishing simplicial volume when the entire fundamental group is amenable.

Results apply to manifolds with multiple ends and simple connectivity at infinity.

Abstract

We show that for $n \neq 1,4$ the simplicial volume of an inward tame triangulable open $n$-manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi_1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.