Averaged Dehn Functions for Nilpotent Groups

TL;DR

This paper confirms Gromov's conjecture that the averaged Dehn function in many nilpotent groups is subasymptotic to the classical Dehn function, with specific bounds related to isoperimetric inequalities.

Contribution

It establishes the relationship between Dehn functions and their averaged versions in nilpotent groups, confirming Gromov's conjecture for most cases and providing sharp bounds for free nilpotent groups.

Findings

Averaged Dehn functions are subasymptotic to Dehn functions in most nilpotent groups.

For groups with isoperimetric inequality elta(l)<Cl^lpha, the averaged inequality is elta^{avg}(l)<C'l^{lpha/2}.

Bounds are asymptotically sharp for non-abelian free nilpotent groups.

Abstract

Gromov proposed an averaged version of the Dehn function and claimed that in many cases it should be subasymptotic to the Dehn function. Using results on random walks in nilpotent groups, we confirm this claim for most nilpotent groups. In particular, if a nilpotent group satisfies the isoperimetric inequality $\delta(l)<Cl^\alpha$ for $\alpha>2$ then it satisfies the averaged isoperimetric inequality $\delta^{\text{avg}}(l)<C'l^{\alpha/2}$. In the case of non-abelian free nilpotent groups, the bounds we give are asymptotically sharp.