Smooth (non)rigidity of piecewise rank one locally symmetric manifolds

Details Smooth (non)rigidity of piecewise rank one locally symmetric manifolds Smooth (non)rigidity of piecewise rank one locally symmetric manifolds

2011-05-26

arxiv.org/abs/1105.5205v2

Mathematics

Geometric Topology

20 pages, 1 figure

Scientific paper

We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with $\pi_1$-injective cusps. We prove smooth (self) rigidity for this class of manifolds in the case where the gluing preserves the cusps' homogeneous structure. We compute the group of self homotopy equivalences of such a manifold and show that it can contain a normal free abelian subgroup and thus, can be infinite. Elements of this abelian subgroup are twists along elements in the center of the fundamental group of a cusp.

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