Special subvarieties of non-arithmetic ball quotients and Hodge Theory

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

Simplicity of some twin tree automorphism groups with trivial commutation relations

We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).Comment: 10 page

Cocompact lattices on A~_n buildings

We construct cocompact lattices Γ’<sub>0</sub>< Γ<sub>0</sub> in the group G = PGL<sub>d</sub>(F<sub>q</sub>((t))) which are type-preserving and act transitively on the set of vertices of each type in the building Δ associated to G. These lattices are commensurable with the lattices of Cartwright [Steger [CS]. The stabiliser of each vertex in Γ’<sub>0</sub> is a Singer cycle and the stabiliser of each vertex in Γ<sub>0</sub> is isomorphic to the normaliser of a Singer cycle in PGL<sub>d</sub>(q). We show that the intersections of Γ’<sub>0</sub> and Γ<sub>0</sub> with PSL<sub>d</sub>(F<sub>q</sub>((t))) are lattices in PSL<sub>d</sub>(F<sub>q</sub>((t))), and identify the pairs (d; q) such that the entire lattice Γ’<sub>0</sub> or Γ<sub>0</sub> is contained in PSL<sub>d</sub>(F<sub>q</sub>((t))). Finally we discuss minimality of covolumes of cocompact lattices in SL<sub>3</sub>(F<sub>q</sub>((t))). Our proofs combine the construction of Cartwright{Steger [CS] with results about Singer cycles and their normalisers, and geometric arguments