Ordered groups

A group G is left ordered if it has a total order <= that is left invariant, so x <= y implies gx <= gy for all g,x,y in G. In the case G is countable, this is equivalent to G being isomorphic to a subgroup of Aut(R), the orientation preserving homeomorphisms of the real line. These two seemingly different ways of viewing a left ordered group can be fruitful. If in addition the total order is right invariant, then G is said to be bi-ordered. I will start by reviewing these definitions, and then discuss some results with some sketch proofs.