On the L^p-distorsion of finite quotients of amenable groups

TL;DR

This paper investigates how the L^p-distortion of finite quotients of amenable groups behaves, showing it grows like a power of log n for lamplighter groups and analyzing other specific groups.

Contribution

It provides new asymptotic results on the L^p-distortion of finite quotients of various classes of amenable groups, including lamplighter, metabelian polycyclic, and Baumslag-Solitar groups.

Findings

L^p-distortion of lamplighter groups grows like (log n)^{1/p}

Asymptotic behavior characterized for certain metabelian polycyclic groups

Results are obtained with short and elementary proofs

Abstract

We study the L^p-distortion of finite quotients of amenable groups. In particular, for every number p larger or equal than 2, we prove that the l^p-distortion of the finite lamplighter group grows like (\log n)^{1/p}. We also give the asymptotic behavior of the l^p-distortion of finite quotients of certain metabelian polycyclic groups and of the solvable Baumslag-Solitar groups BS(m,1). The proofs are short and elementary.