mg.metric geometry - How far can the analogy between a Cayley graph and a symmetric space be pushed?

The Cheeger constants for graphs and Riemannian locally symmetric spaces are closely related. Via inequalities of Buser and Cheeger, these are also related to eigenvalues of the laplacians for each. This analogy led to the first construction of expander graphs, by Margulis, via Property (T). More recently, this analogy has been exploited by several people, notably Marc Lackenby, to study finite-sheeted coverings using Cayley graphs of finite quotients as a finite simplicial approximation.

The point, roughly, is the following. Let $Gamma$ be a group with generating set $S$, and suppose $Gamma = pi_1(M)$ for some Riemannian manifold $M$. Then any finite quotient $F$ under a homomorphism $phi$ has a generating set $phi(S)$, so we can form the corresponding Cayley graph $mathcal{G}(F, phi(S))$. Properties of $mathcal{G}(F, phi(S))$ like girth, spectrum, expansion constants, Cheeger constant, and so forth are closely related to the analogous concept for the finite-sheeted covering $M_phi$ of $M$ corresponding to the subgroup $mathrm{kernel}(phi)$ of $Gamma$. This analogy is most potent when you consider a family {$mathcal{G}(F_j, phi_j(S))$} of Cayley graphs corresponding to a family $F_j$ of finite quotients of $Gamma$.

References for all these concepts are the books On Property ($tau$) by Lubotzky and Zuk (unpublished, but on Lubotzky's website), Discrete Groups, Expanding Graphs and Invariant Measures by Lubotzky, Elementary Number Theory, Group Theory and Ramanujan graphs by Davidoff, Sarnak, and Valette, and Marc Lackenby's paper Expanders, ranks and graphs of groups, Israel J. Math. 146 (2005) 357-370.

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