A few remarks on the octopus inequality and Aldous' spectral gap conjecture

A conjecture by D. Aldous, which can be formulated as a statement about the first nontrivial eigenvalue of the Laplacian of certain Cayley graphs on the symmetric group generated by transpositions, has been recently proven by Caputo, Liggett and Richthammer. Their proof is a subtle combination of two ingredients: a nonlinear mapping in the group algebra of the symmetric groups which permits a proof by induction, and a quite hard estimate named the octopus inequality. In this paper we present a simpler and more transparent proof of the octopus inequality, which emerges naturally when looking at the Aldous' conjecture from an algebraic perspective. We also show that the analogous of the Aldous' conjecture, where the spectral gap is replaced by the Kazhdan constant, does not hold in general.