Commentarii Mathematici Helvetici

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Volume 87, Issue 4, 2012, pp. 789–804
DOI: 10.4171/CMH/268

Published online: 2012-10-10

Smooth (non)rigidity of cusp-decomposable manifolds

T. Tâm Nguyên Phan[1]

(1) University of Chicago, USA

We define cusp-decomposable manifolds and prove smooth rigidity within this class of manifolds. These manifolds generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite-volume, locally symmetric, negatively curved manifolds with cusps. We prove that the group of outer automorphisms of the fundamental group of such manifolds contains a free abelian normal subgroup whose elements are induced by diffeomorphisms that are analogous to Dehn twists in surface topology. For the case where the decomposition is finite, the group of outer automorphisms of the fundamental group is an extension of a finitely generated free abelian group by a finite group. We also prove that the outer automorphism group can be realized by a group of diffeomorphisms of the manifold.

Keywords: Rigidity, outer automorphisms, piecewise locally symmetric manifolds

Nguyên Phan T. Tâm: Smooth (non)rigidity of cusp-decomposable manifolds. Comment. Math. Helv. 87 (2012), 789-804. doi: 10.4171/CMH/268