On intersections of normal subgroups in free groups

Abstract

Let N₁ (respectively N₂) be a normal closure of a set R₁ = {ui} (respectively R₂ = {vj}) of cyclically reduced words of the free group F(A). In the paper we consider geometric conditions on R₁ and R₂ for N₁ ∩ N₂ = [N₁, N₂]. In particular, it turns out that if a presentation < A | R₁, R₂ > is aspherical (for example, it satisfies small cancellation conditions C(p)&T(q) with 1/p + 1/q = 1/2), then the equality N₁ ∩ N₂ = [N₁, N₂] holds.