Subgraph Entropy

2025/06/23 by Aougab, Tarik, Clay, Matt, Hamed, Tawfiq
#Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2506.18838

Abstract

Given r ≥ 3, we prove that there exists λ>0 depending only on r so that if G is a metric graph of rank r with metric entropy 1, then there exists a proper subgraph H of G with metric entropy at least λ. This answers a question of the second two authors together with Rieck. We interpret this as a graph theoretic version of the Bers Lemma from hyperbolic geometry, and explain some connections to the pressure metric on the Culler-Vogtmann Outer Space.

Citations

• Thermodynamic metrics on outer space
• Hyperbolicity of the complex of free factors
• A Bers-like proof of the existence of train tracks for free group automorphisms
• The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores
• Geometry of the complex of curves II: Hierarchical structure
• Extremal length estimates and product regions in Teichmüller space

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