Separation profile, isoperimetry, growth and compression

Abstract

We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Timár) for various graphs using isoperimetric profile, volume growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile is bounded between n a and n b for some a,b∈(0,1). For many amenable groups, we prove a lower bound of n/log(n) a for some a>1, and for groups admitting “good” embeddings into an ℓ p space we prove an upper bound of n/log(n) b for some b∈(0,1). We show that solvable groups of exponential growth cannot have a separation profile bounded above by a sublinear power function. We also introduce a notion of local separation, with applications for percolation clusters of ℤ d and graphs which have polynomial isoperimetry and growth.