Exact descriptions of F{\o}lner functions and sets on wreath products and Baumslag-Solitar groups

TL;DR

This paper computes exact F{46}lner functions for lamplighter groups and wreath products, providing precise isoperimetric data and describing F{46}lner sets, along with an isoperimetric result for Baumslag-Solitar groups.

Contribution

It offers the first exact calculations of F{46}lner functions for specific wreath products and describes the structure of F{46}lner sets, advancing understanding of isoperimetric properties in these groups.

Findings

Exact F{46}lner functions for lamplighter groups and wreath products.

Description of F{46}lner sets that realize these functions.

An isoperimetric inequality for Baumslag-Solitar group BS(1,2).

Abstract

We calculate the exact values of the F{\o}lner function $\mathrm{F{\o}l}$ of the lamplighter group $\mathbb{Z}\wr\mathbb{Z}/2\mathbb{Z}$ for the standard generating set. More generally, for any finite group $D$ and $n\geq|D|$, we obtain the exact value of $\mathrm{F{\o}l}(n)$ on the wreath product $\mathbb{Z}\wr D$, for a generating set induced by a generator on $\mathbb{Z}$ and the entire group being taken as generators for $D$. We also describe the F{\o}lner sets that give rise to it. F{\o}lner functions encode the isoperimetric properties of amenable groups and have previously been studied up to asymptotic equivalence (that is to say, independently of the choice of finite generating set). What is more, we prove an isoperimetric result concerning the edge boundary on the Baumslag-Solitar group $BS(1,2)$ for the standard generating set.