Short incompressible graphs and $2$-free groups

Abstract

Consider a finite connected $2$-complex $X$ endowed with a piecewise Riemannian metric, and whose fundamental group is freely indecomposable, of rank at least $3$, and in which every $2$-generated subgroup is free. In this paper, we show that we can always find a connected graph $\Gamma \subset X$ such that ${\pi }_1\Gamma \simeq {\mathbb{F}}_2\hookrightarrow {\pi }_{1}X$ (in short, a $2$-incompressible graph) whose length satisfies the following curvature-free inequality: $\ell (\Gamma )\le 4\sqrt{2\text{Area}(X)}$. This generalizes a previous inequality proved by Gromov for closed Riemannian surfaces with negative Euler characteristic. As a consequence, we obtain that the volume entropy of such $2$-complexes with unit area is always bounded away from zero.