Separation profiles, isoperimetry, growth and compression

TL;DR

This paper establishes bounds on the separation profile of graphs using isoperimetric, growth, and compression properties, revealing limitations for certain groups and applications to percolation clusters.

Contribution

It provides new bounds for the separation profile based on isoperimetry and growth, and introduces local separation with applications to percolation clusters.

Findings

Separation profile bounds for graphs with polynomial isoperimetry and growth

Lower bounds for amenable groups involving logarithmic factors

Exponential growth solvable groups cannot have sublinear separation profiles

Abstract

We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Tim\'ar) for various graphs using the isoperimetric profile, growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile $\mathrm{Sep}(n)$ is also bounded by powers of $n$. For many amenable groups, we show a lower bound in $n/ \log(n)^a$ and, for any group which has a non-trivial compression exponent in an $L^p$-space, an upper bound in $n/ \log(n)^b$. We show that solvable groups of exponential growth cannot have a separation profile bounded above by a sublinear power function. In an appendix, we introduce the notion of local separation, with applications for percolation clusters of $ \mathbb{Z}^{d} $ and graphs which have polynomial isoperimetry and growth.