Smooth (non)rigidity of cusp-decomposable manifolds (Q1760051)

Given a smooth manifold \(M\), there exists a natural homomorphism \[ \eta :\text{ Diff}(M)/\text{Diff}_0(M) \rightarrow \text{Out}(\pi_1 (M)). \] Where \(\text{Out}(\pi_1 (M))\) is the group of outer automorphisms of the fundamental group \(\pi_1(M)\). It has been proved that \(\eta\) is an isomorphism for closed surfaces (Dehn-Nielson-Baer) and infra-nil manifolds (Auslander). It has also been proved for finite-volume, complete, irreducible, locally symmetric, non-positively curved manifolds of dimension greater than 2 (Mostow). If \(\text{Diff}(M)\) is replaced by the group of homomorphisms \(\Hom(M)\) then the condition holds for closed, non-positively curved manifolds of dimension greater than 4 (Farrel and Jones) and also for solvmanifolds (Mostow). The present author proves that the homomorphism \(\eta\) is an isomorphism for a class of manifolds that are called cusp-decomposable. He introduces cusp-decomposable manifolds \(M\) as obtained by taking finitely many bounded-cusp manifolds \(M_i\) with horoboundary, and glueing them along pairs of boundary components via affine diffeomorphisms. Each pair of boundary components that are glued together can belong to the same \(M_i\). The set of spaces \(M_i\) with the gluing is called a cusp-decomposition of the manifold \(M\). Each \(M_i\) is a piece in the cusp decomposition. A cusp-decomposable manifold \(M\) is of finite type if there are finitely many pieces in the cusp decomposition of M. For any 3-dimensional cusp-decomposable manifold \(M\), there exists a collection of embedded, codimension 1, incompressible, infra-nil submanifolds of \(M\) such that if \(M\) is cut along these submanifolds, the resulting space is a disjoint union of connected manifolds, each of which is diffeomorphic to a finite-volume, non-compact, complete, locally symmetric manifold of negative curvature. For the 2-dimensional case there is also a 2-dimensional analog of the cusp decomposition given by the pair of pants decomposition of a surface.