Short incompressible graphs and $2$-free groups

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 Γ⊂ X such that π₁ Γ F₂π₁ X (in short, a 2 -incompressible graph) whose length satisfies the following curvature-free inequality: (Γ)≤ 4√{2 {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.