  A vertex-transitive graph $\mathcal{G}$ is called Local-to-Global rigid if
there exists $R>0$ such that every other graph whose balls of radius $R$ are
isometric to the balls of radius $R$ in $\mathcal{G}$ is covered by
$\mathcal{G}$. An example of such a graph is given by the Bruhat-Tits building
of $PSL_n(\mathbb{K})$ with $n\geq 4$ and $\mathbb{K}$ a non-Archimedean local
field of characteristic zero.. In this paper we extend this rigidity property
to a class of graphs quasi-isometric to the building including torsion-free
lattices of $SL_n(\mathbb{K})$. The demonstration is the occasion to prove a
result on the local structure of the building. We show that if we fix a
$PSL_n(\mathbb{K})$-orbit in it, then a vertex is uniquely determined by the
neighbouring vertices in this orbit.
