$n$-Kazhdan groups and higher spectral expanders

Let $\Gamma$ be a group of type $F_n$ and let $X$ be the $n$ skeleton of the universal cover of a $K(\Gamma,1)$ simplicial complex with finite $n$ skeleton. We show that if $\Gamma$ is strongly $n$-Kazhdan, then for any family of finite index subgroups $\{\Lambda_i\}_i$, the family of simplicial complexes $\{\Lambda_i\backslash X\}_i$ are bounded degree $n$-dimensional spectral expanders. Using this we construct new examples of $2$ dimensional spectral expanders.