  A biperiodic alternating link has an alternating quotient link in the
thickened torus. In this paper, we focus on semi-regular links, a class of
biperiodic alternating links whose hyperbolic structure can be immediately
determined from a corresponding Euclidean tiling. Consequently, we determine
the exact volumes of semi-regular links. We relate their commensurability and
arithmeticity to the corresponding tiling, and assuming a conjecture of Milnor,
we show there exist infinitely many pairwise incommensurable semi-regular links
with the same invariant trace field. We show that only two semi-regular links
have totally geodesic checkerboard surfaces; these two links satisfy the Volume
Density Conjecture. Finally, we give conditions implying that many additional
biperiodic alternating links are hyperbolic and admit a positively oriented,
unimodular geometric triangulation. We also provide sharp upper and lower
volume bounds for these links.
