Super-expanders and warped cones

Abstract

For a Banach space $X$, we show that any family of graphs quasi-isometric to levels of a warped cone $\mathcal O_\Gamma Y$ is an expander with respect to $X$ if and only if the induced $\Gamma$-representation on $L^2(Y;X)$ has a spectral gap. This provides examples of graphs that are an expander with respect to all Banach spaces of non-trivial type.Comment: 15 pages; to appear in Ann. Inst. Fourier; exposition rewritten, main result slightly generalised to accommodate local spectral gap