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Journal of Topology 2009 2(1):193-225; doi:10.1112/jtopol/jtp005
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© 2009 London Mathematical Society

Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume

Clara Löh

Graduiertenkolleg ‘Analytische Topologie und Metageometrie’, Westfälische Wilhelms-Universität, Münster, Germany, clara.loeh@uni-muenster.de

Roman Sauer

University of Chicago, Chicago, USA, sauerr@uni-muenster.de


  Abstract

We study a metric version of the simplicial volume on Riemannian manifolds, the Lipschitz simplicial volume, with applications to degree theorems in mind. We establish a proportionality principle and a product inequality from which we derive an extension of Gromov's volume comparison theorem to products of negatively curved manifolds or locally symmetric spaces of noncompact type. In contrast, we provide vanishing results for the ordinary simplicial volume; for instance, we show that the ordinary simplicial volume of noncompact locally symmetric spaces with finite volume of Q-rank at least 3 is zero.

Received November 6, 2007. Revised August 20, 2008.

2000 Mathematics Subject Classification 53C23 (primary), 53C35 (secondary)

Current address: Westfälische Wilhelms-Universität, Münster, Germany


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