Connected Polish groups with ample generics

TL;DR

This paper constructs the first examples of connected Polish groups with ample generics, demonstrating their existence and embedding properties, and analyzing the topological rank of full groups of certain ergodic equivalence relations.

Contribution

It provides the first examples of connected Polish groups with ample generics and shows their embedding properties and topological rank results.

Findings

Connected Polish groups with ample generics exist.

Full groups of type III ergodic equivalence relations have ample generics.

Full groups of type III ergodic equivalence relations have topological rank 2.

Abstract

In this paper, we give the first examples of connected Polish groups that have ample generics, answering a question of Kechris and Rosendal. We show that any Polish group with ample generics embeds into a connected Polish group with ample generics and that full groups of type III hyperfinite ergodic equivalence relations have ample generics. We also sketch a proof of the following result: the full group of any type III ergodic equivalence relation has topological rank 2.