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A REDUCTION TO THE COMPACT CASE FOR GROUPS DEFINABLE IN O-MINIMAL STRUCTURES

Published online by Cambridge University Press:  17 April 2014

ANNALISA CONVERSANO
ANNALISA CONVERSANO*
Affiliation:
INSTITUTE OF NATURAL AND MATHEMATICAL SCIENCES MASSEY UNIVERSITY, P/BAG 102-904 NSMC AUCKLAND NEW ZEALANDE-mail: a.conversano@massey.ac.nz
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Abstract

Let ${\cal N}\left( G \right)$ be the maximal normal definable torsion-free subgroup of a group G definable in an o-minimal structure M. We prove that the quotient $G/{\cal N}\left( G \right)$ has a maximal definably compact subgroup K, which is definably connected and unique up to conjugation. Moreover, we show that K has a definable torsion-free complement, i.e., there is a definable torsion-free subgroup H such that $G/{\cal N}\left( G \right) = K \cdot H$ and $K\mathop \cap \nolimits^ \,H = \left\{ e \right\}$. It follows that G is definably homeomorphic to $K \times {M^s}$ (with $s = {\rm{dim}}\,G - {\rm{dim}}\,K$), and homotopy equivalent to K. This gives a (definably) topological reduction to the compact case, in analogy with Lie groups.

Keywords

O-minimalitydefinable groupshomotopy type
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 45 - 53
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

REFERENCES

Baro, E., On the o-minimal LS-category.Israel Journal of Mathematics, vol. 185 (2011), no. 1,pp. 61–76.Google Scholar
Baro, E. and Berarducci, A., Topology of definable abelian groups in o-minimal structures.Bulletin of the London Mathematical Society, vol. 44 (2012), no. 3, pp. 473–479.Google Scholar
Berarducci, A., O-minimal spectra, infinitesimal subgroups and cohomology. this Journal, vol. 72 (2007), no. 4, pp. 1177–1193.Google Scholar
Berarducci, A., Cohomology of groups in o-minimal structures: Acyclicity of the infinitesimal subgroup, this Journal, vol. 74 (2009), no. 3, pp. 891–900.Google Scholar
Berarducci, A. and Mamino, M., On the homotopy type of definable groups in an o-minimal structure. Journal of the London Mathematical Society, vol. 83 (2011), pp. 563–586.Google Scholar
Berarducci, A., Mamino, M., and Otero, M., Higher homotopy of groups definable in o-minimal structures. Israel Journal of Mathematics, vol. 180 (2010), pp. 143–161.Google Scholar
Berarducci, A., Otero, M., Peterzil, Y., and Pillay, A., A descending chain condition for groups definable in o-minimal structures. Annals of Pure and Applied Logic, vol. 134 (2005), pp. 303–313.Google Scholar
Conversano, A., On the connections between definable groups in o-minimal structures and real Lie groups: The non-compact case , Ph.D. thesis, University of Siena, 2009.Google Scholar
Conversano, A., Maximal compact subgroups in the o-minimal setting.Journal of Mathematical Logic, vol. 13 (2013), pp. 1–15.Google Scholar
Conversano, A. and Pillay, A., Connected components of definable groups, and o-minimality I. Advances in Mathematics, vol. 231 (2012), pp. 605–623.Google Scholar
Van Den Dries, L., Tame topology and o-minimal structures , LMS Lecture Notes Series, vol. 248, Cambridge University Press, 1998.Google Scholar
Edmundo, M., Solvable groups definable in o-minimal structures. Journal of Pure and Applied Algebra, vol. 185 (2003), pp. 103–145.Google Scholar
Edmundo, M. and Eleftheriou, P., The universal covering homomorphism in o-minimal expansions of groups. Mathematical Logic Quarterly, vol. 53 (2007), pp. 571–582.Google Scholar
Edmundo, M. and Otero, M., Definably compact abelian groups. Journal of Mathematical Logic, vol. 4 (2004), pp. 163–180.Google Scholar
Gorbatsevich, V. V., Onishchik, A. L., and Vinberg, E. B., Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41. Springer-Verlag, Berlin, 1994.Google Scholar
Hrushovski, E., Peterzil, Y., and Pillay, A., Groups, measures, and the NIP. Journal of American Mathematical Society, vol. 21 (2008), pp. 563–596.Google Scholar
Hrushovski, E., Peterzil, Y., and Pillay, A., On central extensions and definably compact groups in o-minimal structures. Journal of Algebra, vol. 327 (2011), pp. 71–106.Google Scholar
Hrushovski, E. and Pillay, A., On NIP and invariant measures. Journal of the European Mathematical Society, vol. 13 (2011), pp. 1005–1061.Google Scholar
Iwasawa, K., On some types of topological groups. Annals of Mathematics, vol. 50 (1949), no. 3, pp. 507–558.Google Scholar
Knapp, A.W., Lie groups: Beyond an introduction, Birkhäuser, Boston, 2002.Google Scholar
Mamino, M., Splitting definably compact groups in o-minimal structures. Journal of Symbolic Logic, vol. 76 (2011), pp. 973–986.Google Scholar
Otero, M.,A survey on groups definable in o-minimal structures, Model theory with applications to algebra and analysis, vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), LMS LNS 350, Cambridge University Press, 2008, pp. 177–206.Google Scholar
Peterzil, Y., Pillay, A., and Starchenko, S., Definably simple groups in o-minimal structures. Transactions of the American Mathematical Society, vol. 352 (2000), pp. 4397–4419.Google Scholar
Peterzil, Y., Pillay, A., and Starchenko, S., Simple algebraic and semialgebraic groups over real closed fields.Transactions of the American Mathematical Society, vol. 352 (2000), pp. 4421–4450.Google Scholar
Peterzil, Y., Pillay, A., and Starchenko, S., Linear groups definable in o-minimal structures. Journal of Algebra, vol. 247 (2002), pp. 1–23.Google Scholar
Peterzil, Y. and Starchenko, S., On torsion-free groups in o-minimal structures. Illinois Journal of Mathematics, vol. 49 (2008), no. 4, pp. 1299–1321.Google Scholar
Peterzil, Y. and Steinhorn, C., Definable compactness and definable subgroups of o-minimal groups. Journal of the London Mathematical Society, vol. 59 (1999), pp. 769–786.Google Scholar
Pillay, A., On groups and fields definable in o-minimal structures. Journal of Pure and Applied Algebra, vol. 53 (1988), pp. 239–255.Google Scholar
Pillay, A., Type-definability, compact Lie groups, and o-minimality. Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 147–162.Google Scholar
Strzebonski, A., Euler characteristic in semialgebric and other o-minimal group. Journal of Pure and Applied Algebra, vol. 6 (1994), pp. 173–201.Google Scholar