GRUSHKO DECOMPOSITION

Abstract. Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all un-free 2-complexes. Our inequality improves the constant in M. Gro-mov’s inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko’s decom-position of the fundamental group. More specifically, we construct a kind of a Reeb space “minimal model ” for X, reminiscent of the “chopping off long fingers ” construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex