On Groups with Finitely Many Conradian Orderings

Abstract

We study the space of left-orderings on groups with (only) finitely many Conradian orderings. We show that, within this class of groups, having an isolated left-ordering is equivalent to having finitely many left-orderings.

Key Words:

• Conradian orderings
• Isolated orderings
• Space of group orderings

2010 Mathematics Subject Classification:

• 06F15
• 20F60

Notes

Recall that a torsion-free Abelian group Γ has rank n if n is the least integer for which Γ embeds into ℚ ⁿ .

A group G is locally indicable if for any finitely generated subgroup H there is a nontrivial group homomorphism from H to the real numbers under addition.

Two actions φ₁: Γ → Homeo₊(ℝ) and φ₂: Γ → Homeo₊(ℝ) are topologically conjugate if there exists ϕ ∈Homeo₊(ℝ) such that ϕ ○ φ₁(g) = φ₂(g) ○ ϕ for all g ∈ Γ.

Communicated by O. Alexander.