Highly-transitive actions of surface groups

Abstract:

A group action is said to be highly-transitive if it is $k$-transitive for every $k\geq 1$. The main result of this thesis is the following:

Main Theorem. The fundamental group of a closed, orientable surface of genus $> 1$ admits a faithful, highly-transitive action on a countably infinite set.

From a topological point of view, finding a faithful, highly-transitive action of a surface group is equivalent to finding an embedding of the surface group into $\operatorname {sym}{\mathbb {Z}}$ with a dense image. In this topological setting, we use methods that were originally developed for densely embedding surface groups in locally compact groups.