$\ell^p$-distortion and $p$-spectral gap of finite regular graphs

We give a lower bound for the $\ell^p$-distortion $c_p(X)$ of finite graphs $X$, depending on the first eigenvalue $\lambda_1^{(p)}(X)$ of the $p$-Laplacian and the maximal displacement of permutations of vertices. For a $k$-regular vertex-transitive graph it takes the form $c_p(X)^{p}\geq diam(X)^{p}\lambda_{1}^{(p)}(X)/2^{p-1}k$. This bound is optimal for expander families and, for $p=2$, it gives the exact value for cycles and hypercubes. As a new application we give a non-trivial lower bound for the $\ell^2$-distortion of a family of Cayley graphs of $SL_n(q)$ ($q$ fixed, $n\geq 2$) with respect to a standard two-element generating set.