I will overview geometric techniques for considering finite groups of low essential dimension using the equivariant minimal model program. In two dimensions, this approach was pioneered by Serre to determine the essential dimension of the alternating group of degree 6 and I extended this to classify all finite groups of essential dimension 2. Using work of Prokhorov on the space Cremona group, I determined the essential dimensions of the alternating and symmetric groups of degree 7, which Beauville extended to (almost) classify finite simple groups of essential dimension 3. I will also discuss how these techniques are connected to Cremona groups and rationality questions.
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