Integral approximation of simplicial volume of graph manifolds

Abstract

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. Theorem 1.2. Let M be an oriented closed connected graph manifold (in the sense of Definition 2.7) with residually finite fundamental group. Then M = M ∞ Z = 0. More generally, let (Γ j ) j∈N be a descending chain of finite index subgroups of π 1 (M ) with trivial intersection and let (M j ) j∈N be the corresponding tower of finite coverings. Then M = lim j→∞ M j Z [π 1 (M ) : Γ j ] = 0.

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

“…Indeed, we can turn every probability measure-preserving action into an essentially-free action just by taking the product of an essentially-free action and the diagonal action. On the other hand, it seems that working with essentially-free actions is rather common in the literature about simplicial volume and its ergodic version, called integral foliated simplicial volume, see, e.g., [Sau02], [Sch05], [FFM12], [LP16], [FLPS16], [FFL19], [CC], [FLMQ]. For this reason, we think that it is better to keep the same setting here, when we deal with ergodic theory applied to bounded cohomology, which is the dual theory of simplicial volume (notice that we are not claiming that our theory is the dual theory of integral foliated simplicial volume).…”

Stable integral simplicial volume of 3‐manifolds

“…If M is prime with infinite fundamental group and M = 0, then M must be a graph manifold (with infinite fundamental group) [35]. Therefore, we obtain [9]…”

“…The main goal of this paper is to show that non‐elliptic prime 3‐manifolds satisfy integral approximation for simplicial volume (Theorem 1) and that reducible 3‐manifolds in general do not (Section 1.3), thereby answering the approximation question in the 3‐dimensional case [9, Question 1.3].…”

“…The following classes of manifolds were already known to satisfy integral approximation for simplicial volume: closed surfaces of positive genus [18, p. 9], closed hyperbolic 3‐manifolds [13, Theorem 1.7], closed aspherical manifolds with amenable residually finite fundamental group [13, Theorem 1.10], compact manifolds where acts ‘non‐trivially’ [7, 8], as well as graph manifolds not covered by [9].…”

Simplicial Volume

“…In particular, all known vanishing results for the stable integral simplicial volume M ∞ Z imply corresponding vanishing results for M ∞ (Fp) . This includes, for example, the case of aspherical manifolds with residually finite amenable fundamental group [4], smooth aspherical manifolds with nontrivial S 1 -action [2], and graph manifolds [3]. Moreover, also aspherical manifolds with small enough amenable covers have vanishing stable weightless simplicial volume [14] (Example 4.12).…”