An example of a 𝐶-minimal group which is not abelian-by-finite

Simonetta, Patrick

doi:10.1090/S0002-9939-03-06969-7

An example of a $C$-minimal group which is not abelian-by-finite
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Proc. Amer. Math. Soc. 131 (2003), 3913-3917 Request permission

Abstract:

In 1996 D. Macpherson and C. Steinhorn introduced $C$-minimality as an analogue, for valued fields and some groups with a definable chain of normal subgroups with trivial intersection, of the notion of o-minimality. One of the open questions of that paper was the existence of a non abelian-by-finite $C$-minimal group. We give here the first example of such a group.

References

Deirdre Haskell and Dugald Macpherson, Cell decompositions of $\textrm {C}$-minimal structures, Ann. Pure Appl. Logic 66 (1994), no. 2, 113–162. MR 1262433, DOI 10.1016/0168-0072(94)90064-7
D. Haskell, E. Hrushovski, H. D. Macpherson, Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries, submitted.
Alexandre A. Ivanov and Dugald Macpherson, Strongly determined types, Ann. Pure Appl. Logic 99 (1999), no. 1-3, 197–230. MR 1708152, DOI 10.1016/S0168-0072(99)00004-4
L. Lipshitz and Z. Robinson, One-dimensional fibers of rigid subanalytic sets, J. Symbolic Logic 63 (1998), no. 1, 83–88. MR 1610778, DOI 10.2307/2586589
Dugald Macpherson and Charles Steinhorn, On variants of o-minimality, Ann. Pure Appl. Logic 79 (1996), no. 2, 165–209. MR 1396850, DOI 10.1016/0168-0072(95)00037-2
Paulo Ribenboim, The theory of classical valuations, Springer Monographs in Mathematics, Springer-Verlag, New York, 1999. MR 1677964, DOI 10.1007/978-1-4612-0551-7
Patrick Simonetta, Abelian $C$-minimal groups, Ann. Pure Appl. Logic 110 (2001), no. 1-3, 1–22. MR 1846758, DOI 10.1016/S0168-0072(00)00059-2
P. Simonetta, On non-abelian $C$-minimal groups, to appear in Annals of Pure and Applied Logic.