On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ∉ S(a) for all 1 ≠ a ∈ G,then ⟨a ^G ⟩/⟨b ^G ⟩ is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.

Key Words:

• Solvable
• Finite rank
• Orderable groups

Mathematics Subject Classification 2000:

• Primary: 20F16
• Secondary: 20F60
• 06F15

Acknowledgments

Notes

^aThe answer is no. In a paper by Bludov,Glass and Rhemtulla ([2003]) an example of a finitely generated orderable metabelian group G is constructed. This group is residually torsion-free nilpotent but not of finite rank. G/C is nilpotent for every normal non-trivial relatively convex subgroup C of G. It follows from these properties that G is an R ⁺ group.