Simplicial Volume

Abstract

In the past chapters, we explored the interaction between different views on L²-Betti numbers. We will now consider a similar interaction for simplicial volume. Simplicial volume is a numerical topological invariant of manifolds, measuring the “size” of manifolds in terms of the “number” of singular simplices. Simplicial volume is also related to Riemannian volume and geometric structures on manifolds and therefore is a suitable invariant for certain geometric rigidity phenomena. We quickly survey basic properties of simplicial volume and its similarities/differences with L²-Betti numbers and related invariants. In particular, we will discuss the residually finite and the dynamical view on simplicial volume. After this survey, we will sketch some of the prototypical arguments for simplicial volume that involve ergodic theory.