Let [Formula: see text] be a Baumslag–Solitar group and let [Formula: see text] be a complex reductive algebraic group with maximal compact subgroup [Formula: see text]. We show that, when [Formula: see text] and [Formula: see text] are relatively prime with distinct absolute values, there is a strong deformation retraction of [Formula: see text] onto [Formula: see text].
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Weakly minimal groups with a new predicate
Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and let [Formula: see text] be an arbitrary subset of the universe [Formula: see text]. We show that all formulas in the expansion [Formula: see text] are equivalent to bounded formulas, and so [Formula: see text] is stable (or NIP) if and only if the [Formula: see text]-induced structure [Formula: see text] on [Formula: see text] is stable (or NIP). We then restrict to the case that [Formula: see text] is a pure abelian group with a weakly minimal theory, and [Formula: see text] is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of [Formula: see text]. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form [Formula: see text]. Most notably, we show that if [Formula: see text] is a weakly minimal additive subgroup of the algebraic numbers, [Formula: see text] is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of [Formula: see text] is a root of unity, then [Formula: see text] is superstable for any [Formula: see text].
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- Award ID(s):
- 1855789
- NSF-PAR ID:
- 10409415
- Date Published:
- Journal Name:
- Journal of Mathematical Logic
- Volume:
- 20
- Issue:
- 02
- ISSN:
- 0219-0613
- Page Range / eLocation ID:
- 2050011
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1142/S0219061320500117
