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If $G$ is a finitely group and $S$ a finite symmetric set of generators, the associated Cayley graph, then $x \mapsto x^{-1}$ gives rise to a geodesic symmetry $i$ at the identity: If $g=s_1^{e_1}\cdots s_k^{e_k}$ with $e_i \in \{\pm 1\}$, then let $i(g):=s_1^{-e_1}\cdots s_k^{-e_k}$.` Translating $i$ via $G$, there is a symmetry at every vertex. For points in the interior of edges, there is likewise a (local) symmetry.

This prompts my question: What concepts of locally symmetric Riemannian spaces can be applied to the study of Cayley graphs?

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I don't think, that the inversion is a symmetry of the Cayley- graph. Two vertices are adjacent in the Cayley-graph, if they differ by a (say) right multiplication with an element in S. After inverting this turns into left multiplication. For example in the free group inversion is not even a quasi-isometry: a^nb and a^n have distance 1, but their inverses have distance 2n+1. So this question makes only sense, if one considers a modified version of the Cayley-graph, where left and right multiplication is allowed. – HenrikRüping Mar 25 '10 at 12:56
Thank you for the correction. What I meant was the symmetry which is locally given by $x \mapsto x^{-1}$; globally it will have the description as I have now given above. – Guntram Mar 25 '10 at 13:52
isn't there now still a problem. A group element g might be expressed in different ways as a product of the generators and the map might give different results depending on which choice you make, e.g. s1s2=s3, but s^1^-1s_2^-1 \neq s3^-1 – HenrikRüping Mar 26 '10 at 5:51

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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