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Together with the fact that $\\mathrm{SL}(3,\\mathbb{R})$ does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that $\\mathrm{Sp}(2,\\mathbb{R})$ does not have the AP, which was proved by the authors of this article, this finishes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one. 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