Minimal volume entropy of simplicial complexes
Abstract
This article deals with topological assumptions under which the minimal volume entropy of a closed manifold, and more generally of a finite simplicial complex, vanishes or is positive. In the first part of the article, we present complementing topological conditions expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex to simplicial complexes of lower dimension which ensure that the minimal volume entropy of the simplicial complex either vanishes or is positive. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy. In the second part of the article, we present topological assumptions related to the exponential growth of certain subgroups in the fundamental group of a finite simplicial complex and to the topology of the loop space of its classifying space under which the minimal volume entropy is positive. Several examples are presented throughout the text.
Cite
@article{arxiv.2002.11069,
title = {Minimal volume entropy of simplicial complexes},
author = {Ivan Babenko and Stephane Sabourau},
journal= {arXiv preprint arXiv:2002.11069},
year = {2021}
}
Comments
The statement of Theorem 1.1 about the vanishing of the minimal volume entropy under the fiber growth collapsing assumption is incorrect. A correct statement can be found in our preprint entitled "Minimal volume entropy anf fiber growth" corresponding to an extended version of the first part of this paper