In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability
In percolation theory and "random geometry" one is often concerned with amenable graphs, i.e. those with Cheeger constant $0$.
Are these two notions of amenability related?
Yes. Let $G$ be a finitely-generated discrete group with finite generating set $S$. Then $G$ is amenable if and only if the Cayley graph of $G$ with respect to $S$ has Cheeger constant 0. This is known as Følner's criterion for amenability.
For example, $\mathbb{Z}\times\mathbb{Z}$ is amenable because the infinite square grid has Cheeger constant zero, and the free group $F_2$ is nonamenable because an infinite $4$-regular tree has positive Cheeger constant.
