Non-aspherical ends and non-positive curvature (Q2790697)

The authors obtain restrictions on the boundary of a compact manifold whose interior admits a complete Riemannian metric of non-positive sectional curvature \(K\). Their main theorem is the following:NEWLINENEWLINENEWLINELet \(M\) be a closed connected \(m\)-manifold that is either infranil or locally symmetric irreducible with \(K \leq 0\) and real rank \(\geq 2\). If \(N\) is a connected \(n\)-manifold with compact connected boundary such that \(\mathrm{Int}N\) admits a complete metric \(g\) with \(K \leq 0\) and \(\pi _1(\partial N)\cong \pi _1(M)\) then:NEWLINENEWLINENEWLINE1. \(\partial N\) is incompressible if eitherNEWLINENEWLINENEWLINE1a) \(\partial N\) has a neighborhood \(U\) in \(N\) such that \(\mathrm{Vol}_g(U \cap \mathrm{Int}N)\) is finite,NEWLINENEWLINENEWLINEorNEWLINENEWLINENEWLINE 1b) \(\pi _1(M)\) has no proper torsion-free quotients and \(n -m = 1\).NEWLINENEWLINENEWLINE2) If \(n-m\geq 3\), then \(\partial N\) is \(\pi _1\)-incompressible, and \(\partial N\) is the total space of a spherical fibration over \(M\).NEWLINENEWLINENEWLINERecall that a closed manifold is \textit{infranil} if it is of the form \(G/\Gamma\), where \(G\) is a connected, simply-connected, nilpotent Lie group, and \(\Gamma\) is a torsion-free lattice in \(G\rtimes C\), where \(C\) is a maximal compact subgroup of \(\mathrm{Aut}(G)\).