Note on the question of Sikora

TL;DR

The paper discusses the topology of the set of left orderings on free abelian groups and free groups, confirming that this set forms a Cantor set for free groups with more than one generator.

Contribution

It confirms the conjecture that the space of left orderings on free groups is a Cantor set, extending previous results from free abelian groups.

Findings

The topology on left orderings of free abelian groups is a Cantor set.

The topology on left orderings of free groups with n>1 generators is also a Cantor set.

The paper references and confirms a related conjecture.

Abstract

A natural topology on the set of left orderings on free abelian groups and free groups $F_n$, $n>1$ has studied in [1]. It has been proven already that in the abelian case the resulted topological space is a Cantor set. There was a conjecture: this is also true for the free group $F_n$ with $n>1$ generators. We point out the article dealing with equivalent questions. The answer is "yes".