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: $\ell(\Gamma)\leq 4\sqrt{2\text{Area}(X)}$. 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.