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.

15 pages; to appear in Ann. Inst. Fourier; exposition rewritten, main result slightly generalised to accommodate local spectral gaps

References in corpus (2)

• Warped cones, (non-)rigidity, and piecewise properties, with a joint appendix with Dawid Kielak
• Straightening warped cones

Cited by in corpus (1)

• Banach property (T) for $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{Z})$ and its applications