Special subvarieties of non-arithmetic ball quotients and Hodge theory

Let Γ ⊂ PU(1,n) be a lattice and S_Γ be the associated ball quotient. We prove that, if S_Γ contains infinitely many maximal complex totally geodesic subvarieties, then Γ is arithmetic. We also prove an Ax--Schanuel Conjecture for S_Γ, similar to the one recently proven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs is to realise S_Γ inside a period domain for polarised integral variations of Hodge structure and interpret totally geodesic subvarieties as unlikely intersections.