Subgraph Entropy

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.