Subgraph Entropy

Abstract

Given $r\ge 3$, we prove that there exists $\hat{I}»>0$ 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 $\hat{I}»$. 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.

16 pages, 4 figures; v2: changes to the discussion regarding connection to Bers' constant, final version to appear in the Bulletin of the LMS