The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.

中文翻译：

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离散子群的算术性

本课程的主题是半单李群的离散子群。我们讨论了一个确保这样一个子群是算术的标准。该标准是与 Sébastien Miquel 的联合工作，扩展了 Selberg 和 Hee Oh 之前的工作，并解决了 Margulis 的一个古老猜想。我们专注于像小组这样的具体例子$\mathrm {SL}(d,{\mathbb {R}})$我们解释了经典工具和新技术如何进入证明：Auslander 投影定理、Bruhat 分解、Mahler 紧致性准则、Borel 密度定理、Borel-Harish-Chandra 有限性定理、Howe-Moore 混合定理、Dani –Margulis 递归定理、Raghunathan–Venkataramana 有限指数子群定理等。