Ball growth at all scales in universal covers of short metric graphs

Let $\mathrm{\Gamma }$ be a connected metric graph of first Betti number $b\ge 2$, and let ${\mathrm{\Gamma }}_b$ be a trivalent graph of the same Betti number with all edges of unit length. We show that if $\text{length}(\mathrm{\Gamma })\le \lambda \text{length}({\mathrm{\Gamma }}_b)$, then the universal cover of $\mathrm{\Gamma }$ contains a vertex around which, \emph{at every radius} $R\ge 0$, the ball is at least as long as the ball of radius $R$ in the trivalent tree whose edges all have length $3\lambda$. In particular, for $\lambda \le \frac{1}{3}$, this ball is at least as long as the ball of the same radius in the universal cover of ${\mathrm{\Gamma }}_b$, with no multiplicative loss. This improves an earlier theorem of the author, which gives the weaker lower bound $(1-3\lambda )$ times the ball in the universal cover of ${\mathrm{\Gamma }}_b$, and only for $\lambda \le \frac{1}{6}$. The proof rests on an elementary comparison principle. Among trees whose vertices have degree at least three and whose edges have length at most $c$, the trivalent tree with all edges of length $c$ has the smallest balls at every radius. As an application, we double the exponential growth rate in the author’s earlier uniform lower bound for the area of balls in the universal cover of small-area surfaces of hyperbolic type, from $\ln 2$ to $2\ln 2$, beyond the hyperbolic rate. We also obtain explicit constants in the positive answer to Guth’s question for surfaces. Finally, we conjecture the sharp, all-scale version of the minimal volume entropy inequality of Kapovich–Nagnibeda and Lim, and show that it would be optimal.