Algebraic periodicity and scissors congruence for translation surfaces

Calta introduced Property X for the purpose of detecting
genus two translation surfaces with the mysterious lattice property.
McMullen looks at a related invariant for the same purpose in his
work on genus two. I will describe joint work with Kariane Calta
where we explain how these invariants are connected, how they behave
for surfaces of higher genus and how they are connected to scissors
congruence and the J-invariant.