Abstract
This work can be thought of as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable normal subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover.
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A. Berarducci, M. J. Edmundo and M. Mamino, Discrete subgroups of locally definable groups, Selecta Mathematica 19 (2013), 719–736. http://www.springerlink.com/openurl.asp?genre=article\&id=doi:10.1007/s00029-013-0123-9
A. Berarducci and M. Mamino, On the homotopy type of definable groups in an o-minimal structure, Journal of the London Mathematical Society 83 (2011), 563–586.
A. Berarducci, Y. Peterzil and A. Pillay, Group covers, o-minimality, and categoricity, Confluentes Mathematici 02 (2010), 473–496.
A. Conversano and M. Mamino, Private communication.
M. J. Edmundo and P. E. Eleftheriou, The universal covering homomorphism in o-minimal expansions of groups, Mathematical Logic Quarterly 53 (2007), 571–582.
P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Mathematica 18 (2012), 885–903.
A. Fornasiero, Groups and rings definable in d-minimal structures, preprint, arXiv:1205.4177, 2012.
E. Hrushovski, Y. Peterzil and A. Pillay, On central extensions and definably compact groups in o-minimal structures, Journal of Algebra 327 (2011), 71–106.
D. Lascar, Ranks and definability in superstable theories, Israel Journal of Mathematics 23 (1976), 53–87.
J. Marikova, Type-definable and invariant groups in o-minimal structures, Journal of Symbolic Logic 72 (2007), 67–80.
Y. Peterzil and C. Steinhorn, Definable compactness and definable subgroups of o-minimal groups, Journal of the London Mathematical Society 59 (1999), 769–786.
A. Pillay, On groups and fields definable in o-minimal structures, Journal of Pure and Applied Algebra 53 (1988), 239–255.
A. Pillay, Geometric Stability Theory, Oxford Logic Guides, Vol. 32, The Clarendon Press, Oxford University Press, New York, 1996.
J. Ramakrishnan, Y. Peterzil and P. Eleftheriou, Interpretable groups are definable, Journal of Mathematical Logic 14 (2014), 47 pages.
K. Tent and M. Ziegler, A Course in Model Theory, Lecture Notes in Logic, Vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012.
F. Wagner, The right angle to look at orthogonal sets, 2013. arXiv:1310.6275.
R. Wencel, Groups, group actions and fields definable in first-order topological structures, Mathematical Logic Quarterly 58 (2012), 449–467.
B. Zilber, Covers of the multiplicative group of an algebraically closed field of characteristic zero, Journal of the London Mathematical Society 74 (2006), 41–58.
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Partially supported by PRIN 2009WY32E8_003 “O-minimalità, teoria degli insiemi, metodi e modelli non standard e applicazioni”.
Partially supported by Fundação para a Ciência e a Tecnologia grant SFRH/BPD/73859/2010. We also acknowledge the support of FIRB2010, “New advances in the Model Theory of exponentiation”.
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Berarducci, A., Mamino, M. Groups definable in two orthogonal sorts. Isr. J. Math. 208, 413–441 (2015). https://doi.org/10.1007/s11856-015-1205-5
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DOI: https://doi.org/10.1007/s11856-015-1205-5