Local-to-Global-rigidity of lattices in $SL_n(\mathbb{K})$

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.