An $\ell_1$-norm-mass inequality for complete manifolds

TL;DR

This paper extends a classical inequality relating volume and simplicial volume to the $\, ext{ extlbrackdbl} ext{ extperiodcentered} ext{ extrbrackdbl}$-norm of homology classes in complete manifolds, involving the fundamental group's critical exponent.

Contribution

It generalizes the Besson-Courtois-Gallot inequality to include the $\, ext{ extlbrackdbl} ext{ extperiodcentered} ext{ extrbrackdbl}$-norm of all homology classes in complete manifolds, linking geometric and algebraic invariants.

Findings

Established an $\, ext{ extlbrackdbl} ext{ extperiodcentered} ext{ extrbrackdbl}$-norm inequality for complete manifolds.

Connected the critical exponent of the fundamental group to homology class mass.

Extended classical volume inequalities to a broader homological context.

Abstract

We generalize an inequality of Besson-Courtois-Gallot about volume and simplicial volume of closed manifolds to the $\ell_1$-norm of all the homology classes of complete manifolds. The inequality involves the critical exponent of the fundamental group of the manifold and the mass of the homology classes.