A smooth complex variety satisfies the Generalized Jacobian Conjecture if allits etale endomorphisms are proper. We study the equivariant version of theconjecture for $\mathbb{Q}$-acyclic surfaces of negative Kodaira dimension andinfinite algebraic groups. We show that it holds for groups other than$\mathbb{C}^*$, and for $\mathbb{C}^*$ we classify counterexamples relatingthem to Belyi-Shabat polynomials. Taking universal covers we get rationalsimply connected $\mathbb{C}^*$-surfaces of negative Kodaira dimension whichadmit non-proper $\mathbb{C}^*$-equivariant etale endomorphisms. We prove that for every integers $r\geq 1$, $k\geq 2$ the$\mathbb{Q}$-acyclic rational hyperplane $u(1+u^r v)=w^k$, which hasfundamental group $\mathbb{Z}/k\mathbb{Z}$ and negative Kodaira dimension,admits families of non-proper etale endomorphisms of arbitrarily high dimensionand degree, whose members remain different after dividing by the action of theautomorphism group by left and right composition.
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机译：如果所有奇特的内同态都是正确的，则光滑的复杂变体可以满足广义雅可比猜想。我们研究了负Kodaira维数和无限代数群的$ \ mathbb {Q} $-非环面的猜想的等变版本。我们证明它适用于$ \ mathbb {C} ^ * $以外的组，而对于$ \ mathbb {C} ^ * $，我们将与Belyi-Shabat多项式相关的反例分类。采取通用掩盖，我们得到负的Kodaira维数的$ \ mathbb {C} ^ * $曲面的合理连接，这允许非适当的$ \ mathbb {C} ^ * $等效变幻的同质。我们证明对于每个整数$ r \ geq 1 $，$ k \ geq 2 $ the $ \ mathbb {Q} $-非循环有理超平面$ u（1 + u ^ rv）= w ^ k $，它有基本组$ \ mathbb {Z} / k \ mathbb {Z} $以及负的Kodaira维数，允许维和高度任意高的非适当etale同质内生族，其成员除以自同构群的作用除以左右成分后仍然保持不同。
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