$n$-Kazhdan groups and higher spectral expanders

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

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