  In this paper we mainly study Calabi-Yau varieties that arise as triple
covers of products of projective lines branched along simple normal crossing
divisors. For some of those families of Calabi-Yau varieties, the period maps
factor through arithmetic quotients of complex hyperbolic balls. We give a
classification of such examples. One of the families was previously studied by
Voisin, Borcea and Rohde. For these ball-type cases, we will show arithmeticity
of the monodromy groups. These ball quotients are all commensurable to ball
quotients in Deligne-Mostow theory. As a byproduct, we prove some
commensurability relations among arithmetic groups in Deligne-Mostow theory.
