Separation profiles, isoperimetry, growth and compression

2019/10/25 by Corentin Le Coz, Coz, Corentin Le, Antoine Gournay +1 · 2 citations
Mathematics · #05C40 #20F16 #20F65 (primary) #20F67 #20F69 (secondary) #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #Probability (math.PR) #math.GR #math.MG #math.PR #msc:05C40 #msc:20F16 #msc:20F65 #msc:20F67 #msc:20F69

paper · pdf · doi:10.48550/arxiv.1910.11733

41 pages

arxiv created 2019/10/25 · arxiv updated 2019/10/28

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 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 ℤ^d and graphs which have polynomial isoperimetry and growth.

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