TL;DR
This paper proves that the space of left-orderings on free products of groups is a Cantor set and that the free group's conjugation action on its space of orderings has a dense orbit, revealing complex topological dynamics.
Contribution
It establishes that no left-ordering on a free product of groups is isolated and provides a new constructive proof of the dense orbit property for free groups.
Findings
The space of left-orderings of free products is homeomorphic to the Cantor set.
No left-ordering on a free product of groups is isolated.
The conjugation action of free groups on their space of orderings has a dense orbit.
Abstract
We show that no left-ordering on a free product of (left-orderable) groups is isolated. In particular, we show that the space of left-orderings of free product of finitely generated groups is homeomorphic to the Cantor set. With the same techniques, we also give a new and constructive proof of the fact that the natural conjugation action of the free group (on two or more generators) on its space of left-orderings has a dense orbit.
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