Separation profiles, isoperimetry, growth and compression

Abstract

We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Timár) 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.

41 pages

References in corpus (1)

• Random walks on supercritical percolation clusters