Amenability Of Cayley graphs Through Use Of Folner\u27s Conditions

Abstract

In this thesis we will study the definitions and properties relating to groups and Cayley graphs, as well as the concept of amenability. We will discuss McMullen\u27s theorem that states that an infinite tree $X,$ with every vertex having degree equal to 2 is amenable, otherwise if every vertex has degree greater than 2, $X$ is nonamenable. We will also examine how if $G$ is a finitely generated group acting on a set $X$, where $A$ and $B$ are two finite symmetric generating sets of $G$, then the Cayley graph $\textsf{Cay}_A(G,X)$ is amenable if and only if $\textsf{Cay}_B(G,X)$ is amenable. We will show that $(G,X)$ satisfies F\o lner\u27s condition if and only if for every finitely generated subgroup $H$ of $G$, $\textsf{Cay}(H,X)$ is amenable. We will prove that for a finitely generated group $G$, $(G,X)$ is amenable if and only if $\textsf{Cay}(G,X)$ is amenable; this is derived from the fact that $(G,X)$ and $\textsf{Cay}(G,X)$ have the same F\o lner\u27s sequences