A new uncountably categorical group

Baudisch, Andreas

doi:10.1090/S0002-9947-96-01623-6

A new uncountably categorical group
HTML articles powered by AMS MathViewer

by Andreas Baudisch PDF

Trans. Amer. Math. Soc. 348 (1996), 3889-3940 Request permission

Abstract:

We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CM-trivial.

References

J. T. Baldwin and A. H. Lachlan, On strongly minimal sets, J. Symbolic Logic 36 (1971), 79–96. MR 286642, DOI 10.2307/2271517
Andreas Baudisch, Decidability and stability of free nilpotent Lie algebras and free nilpotent $p$-groups of finite exponent, Ann. Math. Logic 23 (1982), no. 1, 1–25. MR 674671, DOI 10.1016/0003-4843(82)90008-0
Andreas Baudisch, On Lascar rank in nonmultidimensional $\omega$-stable theories, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 33–51. MR 895637, DOI 10.1016/S0049-237X(09)70546-4
Andreas Baudisch, Classification and interpretation, J. Symbolic Logic 54 (1989), no. 1, 138–159. MR 987328, DOI 10.2307/2275021
Andreas Baudisch, On superstable groups, J. London Math. Soc. (2) 42 (1990), no. 3, 452–464. MR 1087220, DOI 10.1112/jlms/s2-42.3.452
A. Baudisch, A new uncountably categorical pure group, Preprint Nr. A 93-40 Freie Universität Berlin 1993.
O. V. Belegradek, On groups of finite Morley rank, Abstracts of the $8^{th}$ International Congress of Logic, Methodology, and Philosophy of Science, Moscow, 1987, 100.
Ch. Berline and D. Lascar, Superstable groups, Ann. Pure Appl. Logic 30 (1986), no. 1, 1–43. Stability in model theory (Trento, 1984). MR 831435, DOI 10.1016/0168-0072(86)90035-7
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
J. B. Goode, Hrushovski’s geometries, Proceedings of the $7^{th}$ Easter Conference on Model Theory 1989, Seminarberichte der Humboldt-Universität zu Berlin Nr. 104, 106-117.
Claus Grünenwald and Frieder Haug, On stable torsion-free nilpotent groups, Arch. Math. Logic 32 (1993), no. 6, 451–462. MR 1245526, DOI 10.1007/BF01270468
Ehud Hrushovski, A new strongly minimal set, Ann. Pure Appl. Logic 62 (1993), no. 2, 147–166. Stability in model theory, III (Trento, 1991). MR 1226304, DOI 10.1016/0168-0072(93)90171-9
U. Hrushovski and A. Pillay, Weakly normal groups, Logic colloquium ’85 (Orsay, 1985) Stud. Logic Found. Math., vol. 122, North-Holland, Amsterdam, 1987, pp. 233–244. MR 895647, DOI 10.1016/S0049-237X(09)70556-7
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
Michael Morley, Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), 514–538. MR 175782, DOI 10.1090/S0002-9947-1965-0175782-0
Ali Nesin, Poly-separated and $\omega$-stable nilpotent groups, J. Symbolic Logic 56 (1991), no. 2, 694–699. MR 1133096, DOI 10.2307/2274711
Bruno Poizat, Cours de théorie des modèles, Bruno Poizat, Lyon, 1985 (French). Une introduction à la logique mathématique contemporaine. [An introduction to contemporary mathematical logic]. MR 817208
Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987 (French). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR 902156
Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1983), no. 4, 1151–1170 (1984) (French). MR 727805, DOI 10.2307/2273680
Anand Pillay, Some remarks on modular regular types, J. Symbolic Logic 56 (1991), no. 3, 1003–1011. MR 1129162, DOI 10.2307/2275067
A. Pillay The geometry of forking and groups of finite Morley rank, preprint 1994.
S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
B. I. Zil′ber, Groups and rings whose theory is categorical, Fund. Math. 95 (1977), no. 3, 173–188 (Russian, with English summary). MR 441720
B. I. Zil′ber, Some model theory of simple algebraic groups over algebraically closed fields, Colloq. Math. 48 (1984), no. 2, 173–180. MR 758524, DOI 10.4064/cm-48-2-173-180
Gregory L. Cherlin, Totally categorical structures, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 301–306. MR 804687
B. Zil′ber, Uncountably categorical nilpotent groups and Lie algebras, Soviet Math. (Iz. VUZ) 26 (1982), 98 -99.