Highly Transitive Actions of Surface Groups

Abstract

A group action is said to be highly-transitive if it is $k$-transitive for every $k\ge 1$. The main result of this thesis is the following: Main Theorem: The fundamental group of a closed, orientable surface of genus > 1 admits a faithfull, highly-transitive action on a countably infinite set. From a topological point of view, finding a faithfull, highly-transitive action of a surface group is equivalent to finding an embedding of the surface group into $Sym(Z)$ with a dense image. In this topological setting, we use methods originally developed in [3] and [1] for densely embedding surface groups in locally compact groups.

References in corpus (2)

• Dense embeddings of surface groups
• Compactifications and algebraic completions of Limit groups

Cited by in corpus (1)

• Highly Transitive Actions of Out(Fn)