An isoperimetric inequality for conjugation-invariant sets in the symmetric group

Abstract

We prove an isoperimetric inequality for conjugation-invariant sets of size k in S _n , showing that these necessarily have edge-boundary considerably larger than some other sets of size k (provided k is small). Specifically, let T _n denote the Cayley graph on S _n generated by the set of all transpositions. We show that if A ⊂ S _n is a conjugation-invariant set with |A| = pn! ≤ n!/2, then the edge-boundary of A in T _n has size at least

$$c \cdot \frac{{{{\log }_2}\left( {\frac{1}{p}} \right)}}{{{{\log }_2}{{\log }_2}\left( {\frac{2}{p}} \right)}} \cdot n \cdot \left| A \right|$$

, where c is an absolute constant. (This is sharp up to an absolute constant factor, when p = Θ(1/s!) for any s ∈ {1, 2, …,n}.) It follows that if p = n ^-Θ(1), then the edge-boundary of a conjugation-invariant set of measure p is necessarily a factor of Ω(log n/ log log n) larger than the minimum edge-boundary over all sets of measure p.