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Let $\Gamma \subset \operatorname{PU}(1,n)$ be a lattice, and $S_\Gamma$ theassociated ball quotient. We prove that, if $S_\Gamma$ contains infinitely manymaximal totally geodesic subvarieties, then $\Gamma$ is arithmetic. We alsoprove an Ax-Schanuel Conjecture for $S_\Gamma$, similar to the one recentlyproven by Mok, Pila and Tsimerman. One of the main ingredients in the proofs isto realise $S_\Gamma$ inside a period domain for polarised integral variationsof Hodge structures and interpret totally geodesic subvarieties as unlikelyintersections.

Special subvarieties of non-arithmetic ball quotients and Hodge Theory