Essentially periodic ordered groups

A totally ordered group G is essentially periodic if for every denable non-trivial convex subgroup H of G every denable subset of G is equal to a nite union of cosets of subgroups of G on some interval containing an end segment of H ;i t iscoset-minimal if all denable subsets are equal to a nite union of cosets, intersected with intervals. We study denable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek et al. (J. Symbolic Logic, to appear). Main results: An essentially periodic group G is abelian; if G is discrete, then denable functions in one variable are ultimately piecewise linear. A group such that every model elementarily equivalent to it is coset-minimal is quasi-o-minimal (and vice versa), and its denable functions in one variable are piecewise linear. c 2000 Elsevier Science B.V. All rights reserved. MSC: 03G10; 05E20; 20E36