[Paper Review] Minimal volume entropy of simplicial complexes

Ivan Babenko, Stéphane Sabourau|arXiv (Cornell University)|Feb 25, 2020

Topological and Geometric Data Analysis40 references7 citations

TL;DR

This paper establishes topological conditions under which the minimal volume entropy of a finite simplicial complex is either positive or zero, using growth properties of fundamental groups and loop space topology. It proves that if a complex is essential, its fundamental group is thick (in the sense of algebraic entropy), and its loop space is m-tamed, then the minimal volume entropy is bounded below by a positive constant depending only on dimension, resolving a key case where simplicial volume vanishes but minimal volume entropy remains positive.

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.

Motivation & Objective

• To determine topological conditions ensuring that the minimal volume entropy of a finite simplicial complex is positive or vanishes.
• To investigate whether zero simplicial volume implies zero minimal volume entropy, particularly in higher dimensions.
• To generalize prior results on minimal volume entropy using algebraic entropy and loop space topology.
• To construct examples of finite simplicial complexes with zero simplicial volume but arbitrarily large minimal volume entropy.
• To establish a lower bound for minimal volume entropy in terms of the thickness of the fundamental group and m-tameness of the loop space.

Proposed method

• Define the minimal volume entropy as the infimum of ent(X,g) · vol(X,g)^{1/m} over all piecewise Riemannian metrics g on X.
• Use the notion of ϕ-essentiality to relate the fundamental group of X to a target group G via a homomorphism φ: π₁(X) → G.
• Introduce the concept of δ-thick groups, where every exponentially growing finitely generated subgroup has algebraic entropy ≥ δ.
• Apply volume growth estimates in the universal cover using word metrics on Cayley graphs and volume of balls.
• Use the m-tameness condition on the loop space of the classifying space K(G,1) to control the growth of short loops.
• Derive a lower bound for ent(X,g) via the chain ent(H₀) ≤ ent(H) ≤ ent(X,g) · ℓ_φ(X), where H₀ = φ(H) has exponential growth.

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 there a uniform lower bound for minimal volume entropy in terms of algebraic entropy of subgroups of the fundamental group?
• RQ4 How does the m-tameness of the loop space of K(G,1) relate to the minimal volume entropy of X?
• RQ5 Can one construct examples of finite simplicial complexes with zero simplicial volume but arbitrarily large minimal volume entropy?

Key findings

• The minimal volume entropy of a finite simplicial complex is positive if the complex is ϕ-essential, the target group G is δ-thick, and the loop space of K(G,1) is m-tamed.
• A lower bound ω(X) ≥ λₘδ is established for the minimal volume entropy, where λₘ is a positive constant depending only on the dimension m.
• The paper constructs examples of finite simplicial complexes with zero simplicial volume but arbitrarily large minimal volume entropy, showing that simplicial volume does not control minimal volume entropy in general.
• For complexes with fundamental group that is δ-thick (e.g., discrete subgroups of isometries of pinched negatively curved manifolds), the minimal volume entropy is bounded below by a positive constant.
• The minimal volume entropy is bounded below by δ/R when every ball of radius R has volume ≤ aₘR^m, linking local geometry to entropy.
• The result generalizes previous work by showing that minimal volume entropy positivity can be detected via algebraic entropy and loop space topology, even when simplicial volume vanishes.