On the conjugacy problem in group $\bf F/{N_1\cap N_2}$

Abstract

Let $N_1$ (resp., $N_2$) be the normal closure of a finite symmetrized set $R_1$ (resp., $R_2$) of a finitely generated free group $F = F(A)$. It is well-known that if $R_i$ satisfies the condition C(6), then the conjugacy problem is solvable in $F/N_i$. In the present paper we prove that if $R_1\cup R_2$ satisfies the condition C(6) and the presentation $<A\, \mid R_1,R_2>$ is atorical, then the conjugacy problem is solvable in $F/{N_1\cap N_2}$. In particular, if $R_1\cup R_2$ satisfies the condition C(7) then the conjugacy problem is solvable in $F/{N_1\cap N_2}$.