[Paper Review] Minimal volume entropy and fiber growth

Ivan Babenko, Stéphane Sabourau|arXiv (Cornell University)|Feb 8, 2021

Mathematical Dynamics and Fractals43 references4 citations

TL;DR

This paper establishes topological conditions under which the minimal volume entropy of a finite simplicial complex is positive or zero, using fiber growth of fundamental groups in maps to lower-dimensional complexes. It proves the existence of finite simplicial complexes with zero simplicial volume but arbitrarily large minimal volume entropy, providing a negative answer to a long-standing question about the relationship between simplicial volume and minimal volume entropy.

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.

Motivation & Objective

• To determine topological conditions under which the minimal volume entropy of a finite simplicial complex is positive or vanishes.
• To investigate whether zero simplicial volume implies zero minimal volume entropy, a question open for closed manifolds.
• To construct explicit examples of finite simplicial complexes with zero simplicial volume but positive minimal volume entropy.
• To extend the understanding of minimal volume entropy beyond closed manifolds to general finite simplicial complexes.

Proposed method

• Introduces a fiber growth condition on the fundamental group of fibers in maps from a simplicial complex to lower-dimensional complexes.
• Uses the non-collapsing assumption based on uniform exponential growth of fiber fundamental groups to ensure positive minimal volume entropy.
• Applies the comparison principle and volume entropy semi-norm to relate minimal volume entropy of covers and subcomplexes.
• Employs connected sums and cyclic covers to construct complexes with zero simplicial volume but diverging minimal volume entropy.
• Leverages the additivity of simplicial volume under connected sums in dimension ≥3 to show positivity of minimal volume entropy in covers.
• Uses the volume entropy semi-norm and stabilization processes to analyze asymptotic behavior of minimal volume entropy.

Experimental results

Research questions

• RQ1 Under what topological conditions does the minimal volume entropy of a finite simplicial complex vanish or remain positive?
• RQ2 Can a finite simplicial complex have zero simplicial volume yet positive minimal volume entropy?
• RQ3 Is the minimal volume entropy of a closed manifold with zero simplicial volume necessarily zero?
• RQ4 How does the growth of the fundamental group of fibers in a map to a lower-dimensional complex affect the minimal volume entropy?
• RQ5 Can one construct examples of complexes with zero simplicial volume but arbitrarily large minimal volume entropy?

Key findings

• There exist finite simplicial complexes with zero simplicial volume and arbitrarily large minimal volume entropy, answering a key question in the negative.
• The minimal volume entropy of a complex is positive if its fundamental group satisfies a non-collapsing condition tied to uniform exponential growth in fiber subgroups.
• For the constructed complex $X_n = igvee_{i=1}^n X$, the minimal volume entropy satisfies $\omega(X_n)^2 = n \cdot \omega(X)^2$, so it diverges as $n \to \infty$.
• The minimal volume entropy of the connected sum $Z_n = X_n \# \mathbb{T}^m$ satisfies $\omega(Z_n) \geq \omega(X_n)$, and thus also diverges despite $\|Z_n\|_\Delta = 0$.
• The construction generalizes to even dimensions $m=2k$ using connected sums of manifolds with boundary and cyclic group actions.
• The minimal volume entropy of the $d$-sheeted cover $\widehat{X}_n$ satisfies $\omega(\widehat{X}_n) \leq d^{1/m} \omega(X_n)$, and since $\omega(M_n) \to \infty$, so does $\omega(X_n)$.