Smooth (non)rigidity of piecewise cusped locally symmetric manifolds (Q2892850)

The author studies rigidity properties of manifolds obtained by gluing, possibly infinitely many, manifolds with boundary that are compactifications of noncompact, complete, finite volume, irreducible, locally symmetric, nonpositively curved manifolds along their boundaries. They are defined as piecewise cusped locally symmetric manifolds. Examples are also given. These are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with \(\pi1\)-injective cusps.NEWLINENEWLINE The author proves smooth (self)-rigidity results for this class of manifolds in the case where the gluing preserves the cusps homogeneous structure. The fundamental group of piecewise cusped locally symmetric manifolds is studied (with the corresponding underlying graph of the groups structure of the fundamental group). He computes the group of selfhomotopy equivalences of such a manifold and shows that it can contain a normal free abelian subgroup. Elements of this abelian subgroup are twists along elements in the center of the fundamental group of a cusp.NEWLINENEWLINEIn this paper the results are focused only on the case of gluing other cusped locally symmetric manifolds, those with R-rank at least 2 and Q-rank 1.