The Jacobian Conjecture fails for pseudo-planes

We prove also that for every integers r≥1,k≥2 the Q-acyclic rational hyperplane u(1+urv)=wk, which has fundamental group Zk and negative logarithmic Kodaira dimension, admits families of non-proper étale endomorphisms of arbitrarily high dimension and degree, whose members remain different after dividing by the action of the automorphism group by left and right composition.