Universal sequences for the order-automorphisms of the rationals

In this paper, we consider the group Aut(Q, ≤) of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin’s for the symmetric group. In an announcement, Khélif states that every countable subset of Aut(Q, ≤) is contained in an N-generated subgroup of Aut(Q, ≤) for some fixed N ∈ N. We show that the least such N is 2. Moreover, for every countable subset of Aut(Q, ≤), we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that a and b freely generate the free semigroup {a, b}⁺ consisting of the non-empty words over a and b. Then we show that there exists a sequence of words w₁, w₂, … over {a, b} such that for every sequence f₁, f₂, … ∈ Aut(Q, ≤) there is a homomorphism φ: {a, b}⁺ → Aut(Q, ≤) where (w_i)φ = f_i for every i. As a corollary to the main theorem in this paper, we obtain a result of Droste and Holland showing that the strong cofinality of Aut(Q, ≤) is uncountable, or equivalently that Aut(Q, ≤) has uncountable cofinality and Bergman’s property.