arXiv subjects
Publications and source records attributed to Daniel Fauser.
We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, i.e., that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial volume and tools from ergodic theory.
Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of rank gradients, Betti number gradients and torsion homology gradients for graph manifolds.
The simplicial volume of oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. In the present work we prove a version of this result for the integral foliated simplicial volume of aspherical manifolds: The integral foliated simplicial volume of aspherical oriented closed connected smooth manifolds that admit a non-trivial smooth $S^1$-action vanishes. Our proof uses the geometric construction of Yano's proof for ordinary simplicial volume as well as the parametrised uniform boundary condition for $S^1$.
The uniform boundary condition in a normed chain complex asks for a uniform linear bound on fillings of null-homologous cycles. For the $\ell^1$-norm on the singular chain complex, Matsumoto and Morita established a characterisation of the uniform boundary condition in terms of bounded cohomology. In particular, spaces with amenable fundamental group satisfy the uniform boundary condition in every degree. We will give an alternative proof of statements of this type, using geometric F{\o}lner arguments on the chain level instead of passing to the dual cochain complex. These geometric methods have the advantage that they also lead to integral refinements. In particular, we obtain applications in the context of integral foliated simplicial volume.
Functorial semi-norms on singular homology give refined "size" information on singular homology classes. A fundamental example is the l^1-semi-norm. We show that there exist finite functorial semi-norms on singular homology that are exotic in the sense that they are not carried by the l^1-semi-norm.