Universal sequences for the order-automorphisms of the rationals

In this paper, we consider the group Aut$(\mathbb{Q},\le )$ of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin’s for the symmetric group. In an announcement, Kh'elif states that every countable subset of Aut$(\mathbb{Q},\le )$ is contained in an $N$-generated subgroup of Aut$(\mathbb{Q},\le )$ for some fixed $N\in \mathbb{N}$. We show that the least such $N$ is $2$. Moreover, for every countable subset of Aut$(\mathbb{Q},\le )$, 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 $w_1,w_2,\dots$ over $\{a,b\}$ such that for every sequence $f_1,f_2,\dots \in$Aut$(\mathbb{Q},\le )$ there is a homomorphism $\varphi :\{a,b{\}}^{+}\to$ Aut$(\mathbb{Q},\le )$ where $(w_i)\varphi =f_i$ 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$($\mathbb{Q}$, \leq)$isuncountable,orequivalentlythatAut$($\mathbb{Q}$, \leq)$ has uncountable cofinality and Bergman’s property.