A growth dichotomy for o-minimal expansions of ordered groups

Abstract:

Let $\mathfrak {R}$ be an o-minimal expansion of a divisible ordered abelian group $(R,<,+,0,1)$ with a distinguished positive element $1$. Then the following dichotomy holds: Either there is a $0$-definable binary operation $\cdot$ such that $(R,<,+,\cdot ,0,1)$ is an ordered real closed field; or, for every definable function $f:R\to R$ there exists a $0$-definable $\lambda \in \{0\}\cup \operatorname {Aut}(R,+)$ with $\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R$. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure $\mathfrak {M}:=(M,<,\dots )$ there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) $\mathfrak {M}$-definable groups with underlying set $M$.