$n$-Kazhdan groups and higher spectral expanders

TL;DR

This paper establishes a connection between strongly $n$-Kazhdan groups and the construction of higher-dimensional spectral expanders, providing new examples of 2-dimensional spectral expanders through group-theoretic methods.

Contribution

It proves that strongly $n$-Kazhdan groups yield bounded degree $n$-dimensional spectral expanders via their finite index subgroups, introducing new higher-dimensional expander constructions.

Findings

Strongly $n$-Kazhdan groups produce spectral expanders.

Constructs new 2-dimensional spectral expanders.

Links group properties to high-dimensional expansion.

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.