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

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.

Curve Complexes Versus Tits Buildings: Structures and Applications

“…This vanishing result was proved in [114]. In the proof, Proposition 6.9 and Proposition 6.11 are used crucially.…”

Simplicial volume of ℚ–rank one locally symmetric spaces covered by the product of ℝ–rank one symmetric spaces

“…Such a degree theorem is known by Connell and Farb [7; 6] and Löh and Sauer [19] for proper Lipschitz map f with the sectional curvature of N bounded above by 1 and any n-dimensional locally symmetric manifold M of finite volume. Note that they obtained the degree theorem for proper Lipschitz map f by verifying the positivity of the Lipschitz simplicial volume of M .…”

“…Thurston [22] verified that the simplicial volume of complete Riemannian manifolds with pinched negative sectional curvature and finite volume is strictly positive. In contrast, Gromov [12], Löh and Sauer [19] proved that the simplicial volume of open manifolds, which are the Cartesian product of three open manifolds and locally symmetric spaces of Qrank at least 3, vanishes. Löh and Sauer [20] showed that Hilbert modular varieties have positive simplicial volume, which was the first class of examples of open locally symmetric spaces of R-rank at least 2 for which the positivity of simplicial or minimal volume is known.…”

Piecewise straightening and Lipschitz simplicial volume

“…The proportionality principle provides direct connection between Lipschitz simplicial volume and volume, therefore one obtains immediately Löh and Sauer combined this fact for non-positively curved manifolds with the facts that Lipschitz simplicial volume is strictly positive for locally symmetric spaces of non-compact type of finite volume [1,7,10], that there are only finitely many symmetric spaces (with the standard metric) in each dimension and that N C n vol(N ) if Ricci(N ) −(n − 1) and sec(N ) 1 [4,10] to prove the following theorem. Theorem 1.6 (Degree theorem, [3,7,10]).…”

“…Löh and Sauer studied the above invariant in [10] and proved that it may be a proper generalisation of the simplicial volume to the case of complete Riemannian manifolds of finite volume, not necessarily compact. In particular, in the presence of non-positive curvature they proved the proportionality principle and the product inequality.…”