I will talk about two natural notions of convergence for sequences of graphs of bounded degree and their connection to groups and group actions. The first is Benjamini-Schramm convergence, which is strongly related to parameter testing. The second is local-global convergence, introduced by Hatami, Lovasz and Szegedy. I will discuss recent results on roots of graph polynomials (including the chromatic polynomial and Tutte polynomials), the combinatorics of expander graphings, and the geometry of locally symmetric spaces.

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