# Boundary amenability of groups via ultrapowers

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A (countable discrete) group $\Gamma$ acting on a compact space $X$ is said to act \emph{amenably} if there is a continuous net $(\mu_n^x)$ of probability measures indexed by the points of $X$ that are almost invariant under the action of $\Gamma$. For example, $\Gamma$ is amenable if and only if it acts amenably on a one-point space. The protoypical example of a boundary amenable non-amenable group is a non-abelian free group. More generally, if acts properly, isometrically, and transitively on a tree, then $\Gamma$ is boundary amenable. In this talk, I will present a construction of the Stone-Cech compactification of a locally compact space using C*-algebra ultrapowers that allows one to give a slick proof of the aforementioned result. This construction is motivated by the open question as to whether or not Thompson’s group is boundary amenable and I will also discuss the optimistic thought that this construction could be used to settle that problem.