The spectrum of simplicial volume with fixed fundamental group

TL;DR

This paper investigates the range of simplicial volumes for closed manifolds with a fixed fundamental group, revealing gaps at zero and linking these to rationality issues in bounded cohomology, with implications for volume entropy.

Contribution

It establishes the existence of gaps at zero in the simplicial volume spectrum for many groups and connects these gaps to rationality questions in bounded (co)homology.

Findings

Spectrum has a gap at zero for many groups

Links between simplicial volume gaps and bounded cohomology rationality

Implications for volume entropy in dimension 4

Abstract

We study the spectrum of simplicial volume for closed manifolds with fixed fundamental group and relate the gap problem to rationality questions in bounded (co)homology. In particular, we show that in many cases this spectrum has a gap at zero. For such groups, this leads to corresponding gap results for the minimal volume entropy semi-norm and for the minimal volume entropy in dimension 4.