Highly Transitive Actions of Surface Groups

TL;DR

This paper proves that the fundamental group of a closed, orientable surface of genus greater than one can act in a highly transitive and faithful manner on a countably infinite set, extending understanding of surface group actions.

Contribution

It establishes the existence of faithful, highly-transitive actions of surface groups on infinite sets, a novel result linking topology and permutation group actions.

Findings

Surface groups admit faithful, highly-transitive actions.

Embedding surface groups densely into symmetric groups is possible.

Topological methods can be used to construct such dense embeddings.

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.