Relative expanders

Abstract

We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n⊴G$ such that for every finite generating subset $S\subseteq G$, the sequence of finite Cayley graphs $(G/N_n,S)$ does not coarsely embed into any $L^p$-space for $1⩽p<\infty$ (moreover, into any uniformly curved Banach space), and yet admits no weakly embedded expander. The reason why our examples do not coarsely embed is a new phenomenon called relative expansion, which we define in terms of Poincaré inequalities.

24 pages, new title, Theorem 1.3 is new, more details in proofs of Lemma 2.5 and Theorem 7.3, final revised version

References in corpus (2)

• Nonlinear spectral calculus and super-expanders
• Coarse embeddability into Banach spaces

Cited by in corpus (6)

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• Straightening warped cones
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• On the structure of asymptotic expanders
• The coarse Novikov conjecture for extensions of coarsely embeddable groups
• K-theory of the maximal and reduced Roe algebras of metric spaces with A-by-CE coarse fibrations