TY - JOUR

T1 - Universal sequences for the order-automorphisms of the rationals

AU - Hyde, James

AU - Jonusas, Julius

AU - Mitchell, J. D.

AU - Peresse, Y. H.

N1 - This is the accepted version of the following article: J. Hyde, J. Jonušas, J. D. Mitchell, and Y. Péresse, Universal sequences for the order-automorphisms of the rationals, J. London Math. Soc., first published online May 13, 2016 which has been published in final form at doi:10.1112/jlms/jdw015

PY - 2016/5/13

Y1 - 2016/5/13

N2 - In this paper, we consider the group Aut$(\mathbb{Q}, \leq)$ 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$(\mathbb{Q}, \leq)$ is contained in an $N$-generated subgroup of Aut$(\mathbb{Q}, \leq)$ for some fixed $N\in\mathbb{N}$. We show that the least such $N$ is $2$. Moreover, for every countable subset of Aut$(\mathbb{Q}, \leq)$, 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 $w_1, w_2,\ldots$ over $\{a,b\}$ such that for every sequence $f_1, f_2, \ldots\in\,$Aut$(\mathbb{Q}, \leq)$ there is a homomorphism $\phi:\{a,b\}^{+}\to$ Aut$(\mathbb{Q},\leq)$ where $(w_i)\phi=f_i$ 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$(\mathbb{Q}, \leq)$ is uncountable, or equivalently that Aut$(\mathbb{Q}, \leq)$ has uncountable cofinality and Bergman's property.

AB - In this paper, we consider the group Aut$(\mathbb{Q}, \leq)$ 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$(\mathbb{Q}, \leq)$ is contained in an $N$-generated subgroup of Aut$(\mathbb{Q}, \leq)$ for some fixed $N\in\mathbb{N}$. We show that the least such $N$ is $2$. Moreover, for every countable subset of Aut$(\mathbb{Q}, \leq)$, 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 $w_1, w_2,\ldots$ over $\{a,b\}$ such that for every sequence $f_1, f_2, \ldots\in\,$Aut$(\mathbb{Q}, \leq)$ there is a homomorphism $\phi:\{a,b\}^{+}\to$ Aut$(\mathbb{Q},\leq)$ where $(w_i)\phi=f_i$ 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$(\mathbb{Q}, \leq)$ is uncountable, or equivalently that Aut$(\mathbb{Q}, \leq)$ has uncountable cofinality and Bergman's property.

KW - Group Theory

KW - Infinite Combinatorics

U2 - 10.1112/jlms/jdw015

DO - 10.1112/jlms/jdw015

M3 - Article

SN - 1469-7750

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

ER -