Surveying some of the recent developments on approximate subgroups and super-strong approximation for thin groups, we describe the Bourgain-Gamburd method for establishing spectral gaps for finite groups and the proof of the classification of approximate subgroups of semisimple algebraic groups over finite fields. We then give a proof of the super-strong approximation for mod p quotients via random matrix products and a quantitative version of strong approximation. Some applications to the group sieve are also presented. These notes are based on a series of lectures given at the 2013 Groups St Andrews meeting.
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机译：测量近似亚组的最新发展和薄群的超强近似，我们描述了用于建立有限群体的光谱间隙的Bourgain-Gamburd方法，以及在有限领域的半单代数组的近似子组分类的证明。然后，我们通过随机矩阵产品和强度逼近的定量版本给出Mod P Quotment的超强近似的证据。还提出了对组筛子的一些应用。这些票据基于2013年集团集团集团举行的一系列讲座。
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