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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A growth dichotomy
for o-minimal expansions of ordered groups


Authors: Chris Miller and Sergei Starchenko
Journal: Trans. Amer. Math. Soc. 350 (1998), 3505-3521
MSC (1991): Primary 03C99; Secondary 06F20, 12J15, 12L12
MathSciNet review: 1491870
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Abstract | References | Similar Articles | Additional Information

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 $M$.


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Additional Information

Chris Miller
Affiliation: Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
Address at time of publication: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174
Email: miller@math.ohio-state.edu

Sergei Starchenko
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
Address at time of publication: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: starchenko.1@nd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02288-0
PII: S 0002-9947(98)02288-0
Received by editor(s): June 5, 1996
Additional Notes: The first author was supported by NSF Postdoctoral Fellowship No. DMS-9407549.
Article copyright: © Copyright 1998 American Mathematical Society