L p -distortion and p -spectral gap of finite graphs

Abstract

We give a lower bound for the L p -distortion c p ( X ) of finite graphs X, depending on the first eigenvalue λ 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 ⩾ diam ( X ) p λ 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 new applications we give non-trivial lower bounds for the L 2 -distortion for families of Cayley graphs of the finite lamplighter groups C 2 ≀ C n d ( d ⩾ 2 fixed), and for a family of Cayley graphs of SL n ( q ) ( q fixed, n ⩾ 2 ) with respect to a standard two-element generating set. An application to the L 2 -compression of certain box spaces is also given.