Generic rank of Betti map and unlikely intersections

Let $\mathcal{A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb{C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm{an}}$ one can define the Betti map from $\mathcal{A}_{\Delta}$ to $\mathbb{T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal{A}_{\Delta} \rightarrow \Delta$ with $\mathbb{T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal{A}$ is useful to study Diophantine problems, e.g. proving the Geometric Bogomolov Conjecture over characteristic $0$ and studying the relative Manin-Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large enough fibered power (if $X$ satisfies some conditions): it is an important step to prove the bound for the number of rational points on curves [DGH20]. Another application is to answer a question of Andr\'e-Corvaja-Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin-Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.