A monodromy relation for the Cartwright-Steger fibration

Abstract

We study the genus-$19$ Albanese fibration of the Cartwright--Steger surface and its order-three symmetry. Passing to the orbifold quotient of the elliptic base gives a relation among the two handle monodromies and the three Dehn twists about the vanishing cycles. The two real vanishing paths give cycles whose real fixed points lie on different ovals of the invariant fiber. For a cyclic triple cover, we express the intersection of a curve with its image under the deck transformation as a signed crossing count on the quotient. We determine the two local branch values of a degree-$72$ bicanonical pencil near a node and study a symmetric bicanonical pencil. A general pencil also gives a finite degree-$72$ map from a blowup of the surface to the product of the elliptic curve and a projective line. The monodromy translates of the nodal vanishing classes span the first homology of a smooth fiber.