Ordered groups, knots, braids and hyperbolic 3-manifolds

Free groups are bi-orderable.We will discuss explicit orderings of free groups later.• Braid groups are left-orderable [8].But B n is not bi-orderable if n ≥ 3.• Pure braid groups are bi-orderable [24].Our proof used the Artin "combing" technique expressing the pure braid group as a semidirect product of free groups.The result also follows from a more general result that residually torsion-free nilpotent groups are bi-orderable.• Surface groups (that is, fundamental groups of surfaces) are bi-orderable, except the Klein bottle group 〈, b|  2 = b 2 〉 which is only left-orderable, and the projective plane's group Z/ 2Z which is not even left-orderable.Bi-orderability of surface groups was long known for the orientable case and shown in [23] for the non-oriented (hyperbolic) cases, despite claims in the literature [18] that none of the nonorientable surface groups are bi-orderable.Consider the set Homeo + (R) of order-preserving homeomorphisms of the reals.It is a group under composition of homeomorphisms.• Homeo + (R) is left-orderable.For example, take your favourite enumeration Q = { 1 ,  2 , . ..} of the rationals and compare functions ƒ , g : R → R by declaring ƒ ≺ g iff ƒ (  ) < g(  ) at the first  at which ƒ (  ) and g(  ) differ (this will exist if ƒ = g).Moreover, if G is a countable left-orderable group, then G is isomorphic with a subgroup of Homeo + (R).So Homeo + (R) is universal in the sense that it "contains" all countable leftorderable groups.This provides an intimate connection between ordered groups and dynamics.See for example [9].Clearly left-or bi-orderability is inherited by subgroups.By using a lexicographic ordering, one can also see easily that• A direct product of bi-orderable groups is bi-orderable; similar for left-orderable.The following theorem [27] is more difficult.• A free product of bi-orderable groups is bi-orderable; similar for left-orderable.The Lie group SL 2 (R) of 2 × 2 matrices with real entries and determinant 1 has a universal covering group SL 2 (R).It is one of the eight Thurston geometries for 3-manifolds.• SL 2 (R) is left-orderable.This follows because SL 2 (R) embeds in Homeo + (R).Indeed SL 2 (R) acts on the circle by orientation-preserving homeomorphisms.For example it acts on lines through the origin in R 2 .Another action is by fractional linear transformations of R ∪ ∞.Topologically, SL 2 (R) is homeomorphic with S 1 × D 2 , where D 2 is the open unit disk.Its universal cover SL 2 (R) is an infinite cyclic cover and is a group which acts on R by orientation-preserving homeomorphisms. III-2Course n o III-Ordered groups, knots, braids and hyperbolic 3-manifolds Special algebraic properties of ordered groupsProposition 2.1.1.Left-ordered groups G are torsion-free, that is there are no elements of finite order.