Simplicial volume of manifolds with amenable fundamental group at infinity

In 1982, Gromov introduced a homotopy invariant of manifolds called simplicial volume. Although being purely homotopic in nature, this invariant is sensitive to the geometric structures that a (closed) manifold can carry. The vanishing of the simplicial volume in the case of closed manifolds is implied by the amenability of the fundamental group. For open manifolds the situation is different, since an open manifold with amenable fundamental group can have either vanishing or infinite simplicial volume. A finiteness criterion for simplicial volume of open tame manifolds (i.e., homeomorphic to the interior of a compact manifold with boundary) was given by Clara Loeh in her PhD thesis. It implies that, if the "missing" boundary has amenable fundamental group, then the simplicial volume of the interior is finite. We generalize this result to a wider class of manifolds via the fundamental group at infinity, a topological invariant which detects the topology at infinity of an open manifold. In particular, we prove the amenability of the fundamental group at infinity implies the finiteness of the finite simplicial volume.