Volume entropy semi-norm

Abstract

We introduce the volume entropy semi-norm in real homology and show that it satisfies functorial properties similar to the ones of the simplicial volume. Answering a question of M. Gromov, we prove that the volume entropy semi-norm is equivalent to the simplicial volume semi-norm in every dimension. We also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class.

Minimal volume entropy and fiber growth

“…The techniques developed in this article allow us to provide a negative answer for finite simplicial complexes; see Theorem 1.6. The question for closed orientable manifolds remains open despite recent progress made with the introduction of the volume entropy semi-norm; see [6]. This geometric semi-norm in homology measures the minimal volume entropy of a real homology class throughout a stabilization process.…”

“…where ω(a) is the infimum of the minimal relative volume entropy of the maps f : M → X from an orientable connected closed m-pseudomanifold M to X such that f * ([M ]) = a; see [6] for a more precise definition. The volume entropy semi-norm shares similar functorial features with the simplicial volume semi-norm.…”

“…for some positive constants c m and C m depending only on m. Thus, a closed manifold with zero simplicial volume has zero volume entropy semi-norm, but its minimal volume entropy may be nonzero a priori. See [6] for further details.…”

Minimal volume entropy of simplicial complexes

“…The techniques developed in this article allow us to provide a negative answer for finite simplicial complexes; see Theorem 1.5. The question for closed orientable manifolds remains open despite recent progress made with the introduction of the volume entropy semi-norm; see [7]. This geometric semi-norm in homology measures the minimal volume entropy of a real homology class throughout a stabilization process.…”

“…where ω(a) is the infimum of the minimal relative volume entropy of the maps f : M → X from an orientable connected closed m-pseudomanifold M to X such that f * ([M ]) = a; see [7] for a more precise definition. The volume entropy semi-norm shares similar functorial features with the simplicial volume semi-norm.…”