In classifying fake projective planes, Cartwright and Steger found a smooth compact ball quotient with (topological) Euler number 3 and first betti number 2. I will talk on joint work with Catanese, Keum, and Toledo on determining the geometric properties of this surface, particularly as a surface fibered over the Fermat elliptic curve. My talk will stress how we can understand its geometry via (1) its relationship with one of the orbifolds studied by Deligne and Mostow and (2) the commensurability class of arithmetic lattices containing it.
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