On the geometric Mumford-Tate conjecture for subvarieties of Shimura varieties

We study the image of $\mathrm{\ell }$-adic representations attached to subvarieties of Shimura varieties $S{h}_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\mathrm{\ell }$ large enough (depending on the Shimura datum $(G,X)$ and the subvariety), such image contains the ${\mathbb{Z}}_{\mathrm{\ell }}$-points coming from the simply connected cover of the derived subgroup of $G$. This can be regarded as a geometric version of the integral $\mathrm{\ell }$-adic Mumford-Tate conjecture.