Classification of Groups Which Admit a Finite Number of Distinct Right-Orders

Abstract

Tararin has shown that a non-Abelian group G admits a nonzero finite number of distinct right-orders if and only if G is equipped with a Tararin-type series of some length n. Further, a group which has a Tararin-type series of length n admits 2 ⁿ right-orders. It is known that a group has two right-orders if and only if it is torsionfree Abelian of rank 1. Here we completely classify the groups which admit either four or eight right-orders.

Key Words:

• Right-orderable group
• System of convex subgroups
• Torsionfree Abelian group of rank 1

2000 Mathematics Subject Classification:

• Primary 20F60
• Secondary 06F15, 20E15, 20E22

ACKNOWLEDGMENTS

The author was supported by an EPSRC research studentship. The author is grateful to her former research supervisor, P. H. Kropholler, for many helpful and interesting discussions.

Notes

Communicated by D. Macpherson.