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On enveloping semigroups of nilpotent group actions generated by unipotent affine transformations of the torus

On enveloping semigroups of nilpotent group actions generated by unipotent affine transformations of the torus

Let $G$ be a group generated by a set of affine unipotent transformations $T:X \to X$ of the form $T(x) = A x + \alpha $, where $A$ is a lower triangular unipotent matrix, $\alpha $ is a constant vector, and $X$ is a finite-dimensional torus. We show that