Speaker: Dr. Chenglong Yu (YMSC)

Time: April 26, 2024 (Friday)  14:00-15:00

Title: Commensurabilities among lattices in $\mathrm{PU}(1,n)$

Abstract: In simple Lie groups, except the series $\mathrm{PU}(1,n)$ with $n>1$, either all lattices are all arithmetic, or mathematicians constructed infinitely many nonarithmetic lattices. So far there are only finitely many nonarithmetic lattices constructed for $\mathrm{PU}(1, 2)$ and $\mathrm{PU}(1, 3)$ and no examples for $n>3$. One important construction is via monodromy of hypergeometric functions. The discreteness and arithmeticity of those groups are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of those lattices up to conjugation and finite index (commensurability) is not completed. When $n=1$, it is the commensurabilities of hyperbolic triangles. The cases of $n=2$ are almost resolved by Deligne-Mostow and Sauter's commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some period integrals of higher dimensional varieties instead of complex reflection groups. We obtain some commensurability relations and explicit indices for higher $n$ and also give new proofs for existing pairs in $n=2$. This is based on joint work with Zhiwei Zheng.