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 [Formula Presented] such that [Formula Presented] (in short, a 2-incompressible graph) whose length satisfies the following curvature-free inequality: [Formula Presented]. 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.