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Minimal volume entropy and fiber growth

Ivan Babenko, Stéphane Sabourau

Abstract

This article deals with topological assumptions under which the minimal volume entropy of a closed manifold $M$, and more generally of a finite simplicial complex $X$, vanishes or is positive. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex $X$ to lower dimensional simplicial complexes $P$. We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.

Minimal volume entropy and fiber growth

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. These topological conditions are expressed in terms of the growth of the fundamental group of the fibers of maps from a given finite simplicial complex to lower dimensional simplicial complexes . We also give examples of finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy.

Paper Structure

This paper contains 22 sections, 31 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Simplicial complexes with zero minimal volume entropy
  3. Covering collapsing assumption
  4. Connected and non-connected fibers
  5. Construction of a family of piecewise flat metrics
  6. Construction of Lipschitz retractions around each fiber
  7. Deforming arcs into edge-arcs
  8. Edge-loop entropy
  9. Fiber collapsing assumption and zero minimal volume entropy
  10. Examples of manifolds satisfying the fiber collapsing assumption
  11. Fiber collapsing assumption and zero simplicial volume
  12. Collapsing with Ricci curvature bounded below
  13. Simplicial complexes with positive minimal volume entropy
  14. Covering non-collapsing assumption
  15. Examples of thick groups and non-collapsing simplicial complexes
  16. ...and 7 more sections

Key Result

Theorem \oldthetheorem

Let $X$ be a connected finite simplicial $m$-complex satisfying the fiber $\pi_1$-growth collapsing assumption with subexponential growth rate at most $\nu$ onto a simplicial $k$-complex $P$. Suppose that $\nu < \frac{m-k}{m}$. Then $X$ has zero minimal volume entropy, that is,

Figures (1)

  • Figure 1: Construction of $\mathcal{H}$.

Theorems & Definitions (75)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 65 more