Universal sequences for the order-automorphisms of the rationals

Abstract

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 (wi)φ=fi 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.