Symplectic extension graphs and Ribet's lemma

Let $p$ be an odd prime, and let $\rho \text{\hspace{0.17em}⁣}:G\to {\text{GSp}}_{2n}({\mathbb{Z}}_p)$ be an irreducible symplectic residually multiplicity free representation. We define the symplectic extension graph ${\mathrm{\Gamma }}_{\rho }^{\text{GSp}}$ associated with $\rho$ to be the directed graph whose vertices are the residual factors of $\rho$ and a directed edge between two vertices corresponds to a nonsplit extension between the vertices realized by $\rho$. Using the Euclidean Bruhat–Tits building associated with ${\text{GSp}}_{2n}({\text{Q}}_p)$, under some assumptions, we prove the existence of certain paths in ${\mathrm{\Gamma }}_{\rho }^{\text{GSp}}$ and show that ${\mathrm{\Gamma }}_{\rho }^{\text{GSp}}$ is a disjoint union of nonempty strongly connected subgraphs. Moreover, we provide examples to show that the result we obtain is optimal.