Simplicial volume of manifolds with amenable fundamental group at infinity (Q6648698)

The paper under review studies the relationships between Gromov's simplicial volume, the notion of amenability, and the so-called fundamental group at infinity, for some open manifolds which are ``nice at inifity''.\N\NThe main results are Theorem 1 and Theorem 2: they provide topological conditions at infinity under which an open \(n\)-manifold has finite simplicial volume.\N\NThe first states that open \(n\)-manifolds (with \(n\) greater than 4) which are inward tame and with amenable fundamental group at infinity have finite simplicial volume.\N\NThe second result proves that the same conclusion holds for open \(n\)-manifolds (with \(n\) greater than 4) which are simply connected at infinity and with finitely-many ends.\N\NThe main condition for proving these results within the very large setting of open manifolds is the notion of inward tame manifold. A manifold is said to be \textbf {tame} if it is homeomorphic to the interior of a compact manifold with boundary, while it is called \textbf{inward tame} if every of its neighborhood of infinity can be shrunk into a compact subset within it. The main point here is that such manifolds have finitely many ends with well-defined fundamental group at infinity.\N\NOn the other hand, one says that an end of an open manifold has amenable fundamental group at infinity if the fundamental group at infinity of the end is amenable, in the sense of topological groups. \N\NIn order to prove both theorems the author leans on a couple of recent results from [\textit{R. Frigerio} and \textit{M. Moraschini}, Gromov's theory of multicomplexes with applications to bounded cohomology and simplicial volume. Providence, RI: American Mathematical Society (AMS) (2023; Zbl 1523.55001)].