Abstract
We consider an arbitrary o-minimal structure \(\mathcal {M}\) and a definably connected definable group G. The main theorem provides definable real closed fields \(R_1,\ldots ,R_k\) such that G / Z(G) is definably isomorphic to a direct product of definable subgroups of \(\mathrm{GL}_{n_1}(R_1),\ldots ,\mathrm{GL}_{n_k}(R_k)\), where Z(G) denotes the center of G. From this, we derive a Levi decomposition for G and show that [G, G]Z(G) / Z(G) is definable and definably isomorphic to a direct product of semialgebraic linear groups over \(R_1,\ldots ,R_k\).
Similar content being viewed by others
Notes
Not used, but for the sake of clarity worth keeping.
References
Baro, E., Jaligot, E., Otero, M.: Commutators in groups definable in o-minimal structures. Proc. Am. Math. Soc. 140(10), 3629–3643 (2012)
Borel, A.: Linear Algebraic Groups, 2nd edn. Springer, New York (1991)
Burdges, J.: A signalizer functor theorem for groups of finite Morley rank. J. Algebra 274(1), 215–229 (2004)
Cherlin, G.: Good tori in groups of finite Morley rank. J. Group Theory 8(5), 613–621 (2005)
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)
Conversano, A., Pillay, A.: On Levi subgroups and the Levi decomposition for groups definable in \(o\)-minimal structures. Fund. Math. 222(1), 49–62 (2013)
Derakhshan, J., Wagner, F.O.: Nilpotency in groups with chain conditions. Q. J. Math. Oxf. Ser. (2) 48(192), 453–466 (1997)
Dixon, M.R.: Sylow Theory, Formations and Fitting Classes in Locally Finite Groups, Volume 2 of Series in Algebra. World Scientific Publishing Co., Inc., River Edge, NJ (1994)
Doerk, K., Hawkes, T.: Finite Soluble Groups, Volume 4 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin (1992)
Edmundo, M.J.: Solvable groups definable in o-minimal structures. J. Pure Appl. Algebra 185(1–3), 103–145 (2003)
Eleftheriou, P.E., Peterzil, Y., Ramakrishnan, J.: Interpretable groups are definable. J. Math. Log 14(1), 1450002 (2014)
Frécon, O.: Étude des groupes résolubles de rang de Morley fini. Ph. D. thesis, Université de Lyon (2000)
Frécon, O.: Around unipotence in groups of finite Morley rank. J. Group Theory 9(3), 341–359 (2006)
Frécon, O.: Conjugacy of Carter subgroups in groups of finite Morley rank. J. Math. Log. 8(1), 41–92 (2008)
Frécon, O.: Groupes géométriques de rang de Morley fini. J. Inst. Math. Jussieu 7(4), 751–792 (2008)
Frécon, O.: Pseudo-tori and subtame groups of finite Morley rank. J. Group Theory 12(2), 305–315 (2009)
Grothendieck, A.: Esquisse d’un programme. In: Geometric Galois Actions, 1, Volume 242 of London Math. Soc. Lecture Note Ser., pp. 5–48. Cambridge University Press, Cambridge (1997). With an English translation on pp. 243–283
Miller, C., Starchenko, S.: A growth dichotomy for o-minimal expansions of ordered groups. Trans. Am. Math. Soc. 350(9), 3505–3521 (1998)
Otero, M.: A survey on groups definable in o-minimal structures. In: Model Theory with Applications to Algebra and Analysis, vol. 2, Volume 350 of London Math. Soc. Lecture Note Ser., pp. 177–206. Cambridge University Press, Cambridge (2008)
Otero, M.: On divisibility in definable groups. Fund. Math. 202(3), 295–298 (2009)
Otero, M., Peterzil, Y., Pillay, A.: On groups and rings definable in o-minimal expansions of real closed fields. Bull. Lond. Math. Soc. 28(1), 7–14 (1996)
Peterzil, Y., Pillay, A., Starchenko, S.: Definably simple groups in o-minimal structures. Trans. Am. Math. Soc. 352(10), 4397–4419 (2000)
Peterzil, Y., Pillay, A., Starchenko, S.: Simple algebraic and semialgebraic groups over real closed fields. Trans. Am. Math. Soc. 352(10), 4421–4450 (2000)
Peterzil, Y., Pillay, A., Starchenko, S.: Linear groups definable in o-minimal structures. J. Algebra 247(1), 1–23 (2002)
Peterzil, Y., Starchenko, S.: Definable homomorphisms of abelian groups in o-minimal structures. Ann. Pure Appl. Log. 101(1), 1–27 (2000)
Peterzil, Y., Steinhorn, C.: Definable compactness and definable subgroups of o-minimal groups. J. Lond. Math. Soc. (2) 59(3), 769–786 (1999)
Pillay, A.: On groups and fields definable in \(o\)-minimal structures. J. Pure Appl. Algebra 53(3), 239–255 (1988)
Poizat, B.: Groupes Stables. Nur al-Mantiq wal-Ma’rifah [Light of Logic and Knowledge], 2. Bruno Poizat, Lyon (1987). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An Attempt at Reconciling Algebraic Geometry and Mathematical Logic]
van den Dries, L.: Tame Topology and o-Minimal Structures, Volume 248 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1998)
van den Dries, L.: o-Minimal structures and real analytic geometry. In: Current Developments in Mathematics. 1998 (Cambridge, MA), pp. 105–152. International Press, Somerville, MA (1999)
Warfield Jr, R.B.: Nilpotent Groups. Lecture Notes inMathematics, vol. 513. Springer, Berlin (1976)
Wilkie, A.J.: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Am. Math. Soc. 9(4), 1051–1094 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Frécon, O. Linearity of groups definable in o-minimal structures. Sel. Math. New Ser. 23, 1563–1598 (2017). https://doi.org/10.1007/s00029-016-0247-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-016-0247-9