Proving expansion in three steps

@article{Yehudayoff2012ProvingEI,
  title={Proving expansion in three steps},
  author={Amir Yehudayoff},
  journal={SIGACT News},
  year={2012},
  volume={43},
  pages={67-84}
}
This text is meant to be an introduction to a recent strategy introduced by Bourgain and Gamburd (following a work of Helfgott) for proving graph-expansion. The strategy is designed for graphs H that are defined using some underlying group G. The strategy consists of three steps, which, in Sarnak's terminology, correspond to the three steps of a chess game: opening, middle-game and endgame. In the opening, the objective is to prove that the girth of H is logarithmic. In the middle-game, the… 
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