$n$-Kazhdan groups and higher spectral expanders

Abstract

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.

v2: result improved, new result added, introduction rewritten. 14 pages, no figures