Classification of negatively pinched manifolds with amenable fundamental groups (Q858561)

The authors study manifolds of the form \(X/\Gamma\), where \(X\) is a simply-connected complete Riemannian manifold with sectional curvature pinched between two negative constants, and \(\Gamma\) is a discrete torsion-free subgroup of the isometry group of \(\Gamma\). If \(\Gamma\) is amenable, then either \(\Gamma\) stabilizes a bi-infinite geodesic, or else \(\Gamma\) fixes a unique point at infinity. If \(\Gamma\) fixes a unique point \(z\) at infinity, then \(\Gamma\) stabilizes horospheres centred at \(z\) and permutes geodesics asymptotic to \(z\), so that, given a horosphere \(H\), the manifold \(X/\Gamma\) is diffeomorphic to the product of \(H/\Gamma\) with \(\mathbb{R}\). \(H/\Gamma\) as a horosphere quotient is referred, too. By an inframanifold we mean the quotient of a simply-connected nilpotent Lie group \(G\) by the action of a torsion-free discrete subgroup \(\Gamma\) of the semidirect product of \(G\) with a compact subgroup of \(\Aut(G)\). The main result of this paper is a diffeomorphism classification of horosphere quotients. Namely, the authors prove that for any smooth manifold \(N\) the following are equivalent: (1) \(N\) is a horosphere quotient; (2) \(N\) is diffeomorphic to an infranil manifold; (3) \(N\) is the total space of a flat Euclidean vector bundle over a compact infranilmanifold. The proof of the implication (1) \(\Rightarrow\) (3) is the most complicated part of the proof of the above theorem and depends on the collapsing theory of J. Cheeger, K. Fukaya and M. Gromov.