Equations with constraints on the solution in free groups (Q1326003)

Suppose that \(F_ 3 = \langle a,b,c \rangle\) is the free group with generators \(a\), \(b\), \(c\) and let \(\varphi_ 1\) and \(\varphi_ 2\) be endomorphisms, specified by the following equalities: \(\varphi_ 1(b) = \varphi_ 2(a) = 1\), \(\varphi_ 1(a) = a\), \(\varphi_ 2(b) = b\), \(\varphi_ 1(c) = \varphi_ 2(c) = c\). The following result is proved: There does not exist any algorithm that would permit us to determine, given an arbitrary equation \(w(x_ 1,\dots,x_ m,a,b,c) = 1\), whether it has a solution \(x_ 1,\dots,x_ m\) in the group \(F_ 3\) such that \(\varphi_ 1(x_ 1) = 1\) and \(\varphi_ 2(x_ 1) = 1\). Denoting by \(H\) the intersection of the kernels of the endomorphisms \(\varphi_ 1\) and \(\varphi_ 2\), the following corollary is obtained: There does not exist any algorithm that would permit us to determine, given an arbitrary equation \(w(x_ 1,\dots,x_ m,a,b,c) = 1\), whether it has a solution \(x_ 1,\dots,x_ m\) in \(F_ 3\) such that \(x_ 1 \in H\).