Stable groups and expansions of $(\mathbb Z,+,0)$
Volume 242 / 2018
Fundamenta Mathematicae 242 (2018), 267-279
MSC: Primary 03C45; Secondary 03C64.
DOI: 10.4064/fm403-8-2017
Published online: 26 February 2018
Abstract
We prove that if $G$ is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then $G$ is superstable of finite $U$-rank. Combined with recent work of Palacín and Sklinos, this shows that $(\mathbb Z,+,0)$ has no proper stable expansions of finite weight. A corollary is that if $P\subseteq \mathbb Z^n$ is definable in a finite dp-rank expansion of $(\mathbb Z,+,0)$, and $(\mathbb Z,+,0,P)$ is stable, then $P$ is definable in $(\mathbb Z,+,0)$. In particular, this answers a question of Marker on stable expansions of the group of integers by sets definable in Presburger arithmetic.
