Geodesic
On intersections of normal subgroups in free groups
Algebra and discrete mathematics, no. 1 (2003), pp. 36-67.

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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 $$ 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.
Keywords: normal closure of words in free groups, presentations of groups, pictures, mutual commutants, intersection of groups, aspherisity, small cancellation conditions. 
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     author = {O. V. Kulikova},
     title = {On intersections of normal subgroups in free groups},
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     number = {1},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2003_1_a4/}
}
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O. V. Kulikova. On intersections of normal subgroups in free groups. Algebra and discrete mathematics, no. 1 (2003), pp. 36-67. http://geodesic.mathdoc.fr/item/ADM_2003_1_a4/