Higher-dimensional ball quotients and the invariant quintic

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We prove a conjecture made earlier concerning a beautiful algebraic fourfold, a quintic in projective five-space invariant under the Weyl group of typeE 6, to the effect that a certain birational model of this variety is a smooth compactification of a ball quotient. To prove this, we first state and prove a general result which gives a criterion for checking whether a variety of dimensionN≥3 is a (compactification of a) ball quotient. We then go on to identify the group up to commensurability class.