Simplicial volume of manifolds with amenable fundamental group at infinity

We show that for n ≠ 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 π 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.