Abstract : A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its etale endomorphisms are proper. We study the equivariant version of the conjecture for ℚ-acyclic surfaces of negative Kodaira dimension and infinite algebraic groups. We show that it holds for groups other than ℂ∗, and for ℂ∗ we classify counterexamples relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected ℂ∗-surfaces of negative Kodaira dimension which admit non-proper ℂ∗-equivariant etale endomorphisms.
We prove that for every integers r≥1, k≥2 the ℚ-acyclic rational hyperplane u(1+urv)=wk, which has fundamental group ℤ/kℤ and negative Kodaira dimension, admits families of non-proper etale 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.