Exact versus monotone isoperimetric profiles of finitely generated groups

Abstract

Gromov asked whether the exact isoperimetric profile of an infinite finitely generated group is comparable, at the same cardinality, to its greatest nondecreasing minorant. We compare finite sets with their translates and construct orders for which the first $r$ vertices have boundary bounded by a fixed multiple of that of the whole set. Such orders follow from profile summability or from rays in Schreier graphs with a uniform bound on the number using each edge. Among infinite finitely generated groups, we obtain comparison and nested exhaustions for elementary amenable and branch groups, direct products of at least two infinite groups, restricted wreath products with two nontrivial factors, and certain simple topological full groups, including amenable derived groups of interval exchanges. The exhausting $n$-point sets have boundary comparable to the monotone profile at $n$. The ray condition is quasi-isometry invariant and holds for the Sierpiński-carpet group. For polycyclic groups of exponential growth, we refine Pittet's construction to obtain explicit nested sets with $|F_n|=n$ and $|\partial F_n|\asymp n/\log n$, answering his question with exact rather than comparable cardinalities. We also prove that the normalized $L^1$ profile satisfies $j(v)\le C(v/u)j(u)$ for $1\le u\le v$ on every infinite connected bounded-degree vertex-transitive graph. This yields the separation bound asked for by Gournay--Le Coz and Hume's Dirichlet--Neumann comparison at $p=1$. For groups of exponential growth, the exact-to-monotone comparison ratio is $O(\log \log n)$. We answer the first part of Correia's Question 1.3 positively by proving the required dilation regularity, and the second negatively by constructing elementary amenable examples whose inverse compositions fail this regularity for every choice of equivalent representatives.

arXiv comment37 pages, comments most welcome!