In the free group generated by the letters $x,y,z$ suppose that we have a word such that for any one of $x,y,z$ the indices of each occurrence of that letter in our word sum to zero. Suppose further that if we set any one letter to the identity, the remaining word collapses to the identity. Must our original word have been equivalent to the identity?
Help, or directions to relevant reading material would be greatly appreciated.
Consider $[x,y]([x,y]^{-1})^z = (x^{-1}y^{-1}xy)z^{-1}(y^{-1}x^{-1}yx)z$.