On intersections of normal subgroups in free groups

Abstract

Let \(N_1\) (respectively \(N_2\)) be a normal closure of a set \(R_1=\{ u_i \}\) (respectively \(R_2=\{ v_j \}\)) of cyclically reduced words of the free group \(F(A)\). In the paper we consider geometric conditions on \(R_1\) and \(R_2\) for \(N_1\cap N_2=[N_1,N_2].\) In particular, it turns out that if a presentation \(<A\, \mid R_1,R_2>\)  is aspherical (for example, it satisfies small cancellation conditions \(C(p)\& T(q)\) with \(1/p+1/q=1/2\)), then the equality \(N_1\cap N_2=[N_1,N_2]\) holds.