Vanishing Simplicial Volume for Certain Affine Manifolds
نویسنده
چکیده
We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of π1 to have vanishing simplicial volume. This answers a special case of a question due to Lück.
منابع مشابه
Degree Theorems and Lipschitz Simplicial Volume for Non-positively Curved Manifolds of Finite Volume
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 product formula 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 pr...
متن کاملPassivity-Based Stability Analysis and Robust Practical Stabilization of Nonlinear Affine Systems with Non-vanishing Perturbations
This paper presents some analyses about the robust practical stability of a class of nonlinear affine systems in the presence of non-vanishing perturbations based on the passivity concept. The given analyses confirm the robust passivity property of the perturbed nonlinear systems in a certain region. Moreover, robust control laws are designed to guarantee the practical stability of the perturbe...
متن کامل`1-Homology and Simplicial Volume
Introduction A pervasive theme of contemporary mathematics is to explore rigidity phenomena caused by the symbiosis of algebraic topology and Riemannian geometry on manifolds. In this context, the term " rigidity " refers to the astounding fact that certain topological invariants provide obstructions for geometric structures. Consequently , topological invariants of this type serve as interface...
متن کاملOn the Simplicial Volumes of Fiber Bundles
We show that surface bundles over surfaces with base and fiber of genus at least 2 have non-vanishing simplicial volume. The simplicial volume ||M ||, introduced by Gromov [3], is a homotopy invariant which measures the complexity of the fundamental class of an oriented manifold M . It is determined by the classifying map of the universal covering, and tends to be non-zero for large manifolds o...
متن کاملAmenable Covers, Volume and L-betti Numbers of Aspherical Manifolds
We provide a proof for an inequality between volume and LBetti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau’s theory of L-Betti numbers of R-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for ...
متن کامل