Isoperimetric profiles of lamplighter-like groups

TL;DR

This paper investigates the isoperimetric profiles of lamplighter-like groups, establishing sharp bounds for groups with known exponential growth profiles, and explores applications to regular maps between these groups.

Contribution

It provides new sharp bounds for isoperimetric profiles of lamplighter groups over amenable groups, extending previous estimates and applying to halo products.

Findings

Sharp bounds for isoperimetric profiles of lamplighter groups with exponential growth.

Identification of lamplighter subgraphs in lampshuffler groups for optimal bounds.

Applications to the existence of regular maps between such groups.

Abstract

Given a finitely generated amenable group $H$ satisfying some mild assumptions, we relate isoperimetric profiles of the lampshuffler group $\mathsf{Shuffler}(H)=\mathsf{FSym}(H)\rtimes H$ to those of $H$. Our results are sharp for all exponential growth groups for which isoperimetric profiles are known, including Brieussel-Zheng groups. This refines previous estimates obtained by Erschler and Zheng and by Saloff-Coste and Zheng. The most difficult part is to find an optimal upper bound, and our strategy consists in finding suitable lamplighter subgraphs in lampshufflers. This novelty applies more generally for many examples of halo products, a class of groups introduced recently by Genevois and Tessera as a natural generalisation of wreath products. Lastly, we also give applications of our estimates on isoperimetric profiles to the existence problem of regular maps between such groups.