Special subvarieties of non-arithmetic ball quotients and Hodge Theory

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