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In this paper we mainly study Calabi-Yau varieties that arise as triplecovers of products of projective lines branched along simple normal crossingdivisors. For some of those families of Calabi-Yau varieties, the period mapsfactor through arithmetic quotients of complex hyperbolic balls. We give aclassification of such examples. One of the families was previously studied byVoisin, Borcea and Rohde. For these ball-type cases, we will show arithmeticityof the monodromy groups. These ball quotients are all commensurable to ballquotients in Deligne-Mostow theory. As a byproduct, we prove somecommensurability relations among arithmetic groups in Deligne-Mostow theory.

Calabi-Yau Varieties via Cyclic Covers, and Complex Hyperbolic Structures for their Moduli Spaces