Automorphisms fixing a variable of K〈x,y,z〉

We study automorphisms φ of the free associative algebra K〈x,y,z〉 over a field K such that φ(x),φ(y) are linear with respect to x,y and φ(z)=z. We establish a sufficient and necessary condition for the tameness of these automorphisms in the class of all automorphisms fixing z, which gives an algorithm to recognize the wild ones. In particular, we prove that the well-known Anick automorphism is wild in this sense. This class of automorphisms induces tame automorphisms of the polynomial algebra K[x,y,z]. For n>2 the automorphisms of K〈x1,…,xn,z〉 which fix z and are linear in the xis are tame.