## Strongly Dense Subgroups of Semisimple Algebraic Groups

#### Robert GuralnickUniversity of Southern California (USC)

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p$. A nonabelian free subgroup of $G$ is called strongly dense if every noncyclic subgroup is Zariski-dense. We will discuss existence results and applications of the results and methods to expanders as well as to finite groups of Lie type. We will also discuss some open problems. This is based on some joint work with Breuillard, Green, Larsen and Tao.

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