$n$-Kazhdan groups and higher spectral expanders

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