  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.
