A Growth Dichotomy for O-minimal Expansions of Ordered Groups

Let 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 · such that (R, <, +, ·, 0, 1) is an ordered real closed field; or, for every definable function f : R → R there exists a 0-definable λ ∈ {0} ∪ Aut(R, +) with limx→+∞[f(x) − λ(x)] ∈ R. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure M := (M, <, . . . ) there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) M-definable groups with underlying set M . R. Poston showed in [8] that given an o-minimal expansion R of (R, <, +), if multiplication is not definable in R, then for every definable function f : R → R there exist r, c ∈ R such that limx→+∞[f(x) − rx] = c. In this paper, this fact is generalized appropriately for o-minimal expansions of arbitrary ordered groups. We say that an expansion (G, <, ∗, . . . ) of an ordered group (G, <, ∗) is linearly bounded (with respect to ∗) if for each definable function f : G → G there exists a definable λ ∈ End(G, ∗) such that ultimately |f(x)| ≤ λ(x). (Here and throughout, ultimately abbreviates “for all sufficiently large positive arguments”.) We now list the main results of this paper. Let R := (R, <, . . . ) be o-minimal. Theorem A (Growth Dichotomy). Suppose that R is an expansion of an ordered group (R, <, +). Then exactly one of the following holds: (a) R is linearly bounded; (b) R defines a binary operation · such that (R, <, +, ·) is an ordered real closed field. If R is linearly bounded, then for every definable f : R → R there exist c ∈ R and a definable λ ∈ {0} ∪ Aut(R, +) with limx→+∞[f(x)− λ(x)] = c. Theorem B. Suppose that R is a linearly bounded expansion of an ordered group (R, <, +, 0, 1) with 1 > 0. Then every definable endomorphism of (R, +) is 0-definable. If R′ (with underlying set R′) is elementarily equivalent to R, then the ordered division ring of all R′-definable endomorphisms of (R′, +) is canonically isomorphic to the ordered division ring of all R-definable endomorphisms of (R, +). The growth dichotomy imposes some surprising constraints on continuous definable groups with underlying set R. (Here and throughout, all topological notions are taken with respect to the product topologies induced by the order topology.) Received by the editors June 5, 1996. 1991 Mathematics Subject Classification. Primary 03C99; Secondary 06F20, 12J15, 12L12. The first author was supported by NSF Postdoctoral Fellowship No. DMS-9407549. c ©1998 American Mathematical Society