Integral foliated simplicial volume of aspherical manifolds

Abstract

Abstract. Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between simplicial volume and two of its variants -the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.We show that integral foliated simplicial volume is monotone with respect to weak containment of measure preserving actions and yields upper bounds on (integral) homology growth.Using ergodic theory we prove that simplicial volume, integral foliated simplicial volume and stable integral simplicial volume coincide for closed hyperbolic 3-manifolds and closed aspherical manifolds with amenable residually finite fundamental group (being equal to zero in the latter case).However, we show that integral foliated simplicial volume and the classical simplicial volume do not coincide for hyperbolic manifolds of dimension at least 4.

Variations on the theme of the uniform boundary condition

“…Integral foliated simplicial volume of aspherical oriented closed connected surfaces and hyperbolic 3-manifolds coincides with ordinary simplicial volume [13]. However, for higher-dimensional hyperbolic manifolds, integral foliated simplicial volume is uniformly bigger than ordinary simplicial volume [8].…”

“…Our method of proof is related to the Følner filling argument for the vanishing of integral foliated simplicial volume of aspherical oriented closed connected manifolds with amenable fundamental group [8,Section 6]. More precisely, the proof consists of three steps:…”

“…Because Γ is residually finite and finitely generated, α is an essentially free standard Γ-space and[8, Theorem 2.6] …”

Rank gradient versus stable integral simplicial volume

“…As in the case for the rank gradient, it is unknown whether stable integral simplicial volume is independent of the chosen chain of subgroups (with trivial intersection) or not. For aspherical oriented closed connected surfaces, for closed hyperbolic 3-manifolds, for closed Seifert manifolds with infinite fundamental group, and for aspherical closed manifolds with residually finite amenable fundamental group the stable integral simplicial volume coincides with ordinary simplicial volume [5]. However, for closed hyperbolic manifolds of dimension at least 4, stable integral simplicial volume is uniformly bigger than ordinary simplicial volume [4].…”

“…Stable integral simplicial volume is the gradient invariant associated with integral simplicial volume (Section 3). It is known that stable integral simplicial volume yields upper bounds for Betti number gradients and logarithmic torsion homology gradients [5,Theorem 1.6,Theorem 2.6].…”