We study the model theory of "covers" of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is interpretable in M, proving some cases (such as when H is abelian) as well as stating various equivalences. When M is an o-minimal expansion of the reals (so H is a definable Lie group) this is related to Milnor's conjecture [15], and many cases are known. We also prove a strong relative Lω1, ω-categoricity theorem for universal covers of definable Lie groups, and point out some notable differences with the case of covers of complex algebraic groups (studied by Zilber and his students).
@article{CML_2010__2_4_473_0,
author = {Berarducci, Alessandro and Peterzil, Ya{\textquoteright}acov and Pillay, Anand},
title = {Group covers, $o$-minimality, and categoricity},
journal = {Confluentes Mathematici},
pages = {473--496},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {2},
number = {4},
year = {2010},
doi = {10.1142/S1793744210000259},
language = {en},
url = {http://archive.numdam.org/articles/10.1142/S1793744210000259/}
}
TY - JOUR AU - Berarducci, Alessandro AU - Peterzil, Ya’acov AU - Pillay, Anand TI - Group covers, $o$-minimality, and categoricity JO - Confluentes Mathematici PY - 2010 SP - 473 EP - 496 VL - 2 IS - 4 PB - World Scientific Publishing Co Pte Ltd UR - http://archive.numdam.org/articles/10.1142/S1793744210000259/ DO - 10.1142/S1793744210000259 LA - en ID - CML_2010__2_4_473_0 ER -
%0 Journal Article %A Berarducci, Alessandro %A Peterzil, Ya’acov %A Pillay, Anand %T Group covers, $o$-minimality, and categoricity %J Confluentes Mathematici %D 2010 %P 473-496 %V 2 %N 4 %I World Scientific Publishing Co Pte Ltd %U http://archive.numdam.org/articles/10.1142/S1793744210000259/ %R 10.1142/S1793744210000259 %G en %F CML_2010__2_4_473_0
Berarducci, Alessandro; Peterzil, Ya’acov; Pillay, Anand. Group covers, $o$-minimality, and categoricity. Confluentes Mathematici, Tome 2 (2010) no. 4, pp. 473-496. doi : 10.1142/S1793744210000259. http://archive.numdam.org/articles/10.1142/S1793744210000259/
[1] M. Bays, Categoricity results for exponential maps of 1-dimensional algebraic groups, and Schanuel’s conjecture for Powers and the CIP, Ph.D. thesis, University of Oxford, 2009 .
[2] A. Berarducci, J. Symbolic Logic 74, 1177 (2007).
[3] A. Berarducci, J. Symbolic Logic 74, 891 (2009), DOI: 10.2178/jsl/1245158089.
[4] A. Berarducci and M. Mamino, On the homotopy type of definable groups in an o-minimal structure, to appear in J. London Math. Soc .
[5] A. Berarducciet al., Ann. Pure Appl. Logic 134, 303 (2005), DOI: 10.1016/j.apal.2005.01.002.
[6] J. L. Dupont, W. Parry and C. H. Sah, J. Algebra 113, 215 (1988), DOI: 10.1016/0021-8693(88)90191-3.
[7] M. Edmundo, J. Algebra 301, 194 (2006), DOI: 10.1016/j.jalgebra.2005.04.016.
[8] M. Edmundo, G. O. Jones and N. J. Peatfield, Hurewicz theorems for definable groups, Lie groups and their cohomologies, preprint (http://www.ciul.ul.pt/edmundo/), Oct. 13, 2007, 22 pp .
[9] M. Edmundo , G. O. Jones and N. J. Peatfield , Arch. Math. Logic .
[10] M. Edmundo and P. Eleftheriou, Math. Logic Quart. 53, 571 (2007), DOI: 10.1002/malq.200610051.
[11] M. Gavrilovich, Model theory of the universal covering spaces of complex algebraic varieties, Ph.D. thesis, University of Oxford, 2006 .
[12] E. Hrushovski, Y. Peterzil and A. Pillay, J. Amer. Math. Soc. 21, 563 (2008), DOI: 10.1090/S0894-0347-07-00558-9.
[13] E. Hrushovski, Y. Peterzil and A. Pillay, J. Algebra 327, 71 (2011), DOI: 10.1016/j.jalgebra.2010.11.001.
[14] E. Hrushovski and A. Pillay, On NIP and invariant measures, preprint, 29th January 2009 , arXiv:0710.2330v2 .
[15] J. Milnor, Comment. Math. Helvetici 58, 72 (1983), DOI: 10.1007/BF02564625.
[16] M. Otero, Y. Peterzil and A. Pillay, Bull. London Math. Soc. 28, 7 (1996), DOI: 10.1112/blms/28.1.7.
[17] A. Pillay, J. Pure Appl. Algebra 53, 239 (1988), DOI: 10.1016/0022-4049(88)90125-9.
[18] A. Pillay, J. Math. Logic 4, 147 (2004), DOI: 10.1142/S0219061304000346.
[19] C. H. Sah, Comment. Math. Helvetici 61, 308 (1986), DOI: 10.1007/BF02621918.
[20] B. Zilber, J. London Math. Soc. 74, 41 (2006), DOI: 10.1112/S0024610706023349.
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