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).
Alessandro Berarducci 1 ; Ya’acov Peterzil 1 ; Anand Pillay 1
@article{CML_2010__2_4_473_0,
author = {Alessandro Berarducci and Ya{\textquoteright}acov Peterzil and Anand Pillay},
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 = {https://cml.centre-mersenne.org/articles/10.1142/S1793744210000259/}
}
TY - JOUR AU - Alessandro Berarducci AU - Ya’acov Peterzil AU - Anand Pillay 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 - https://cml.centre-mersenne.org/articles/10.1142/S1793744210000259/ DO - 10.1142/S1793744210000259 LA - en ID - CML_2010__2_4_473_0 ER -
%0 Journal Article %A Alessandro Berarducci %A Ya’acov Peterzil %A Anand Pillay %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 https://cml.centre-mersenne.org/articles/10.1142/S1793744210000259/ %R 10.1142/S1793744210000259 %G en %F CML_2010__2_4_473_0
Alessandro Berarducci; Ya’acov Peterzil; Anand Pillay. Group covers, $o$-minimality, and categoricity. Confluentes Mathematici, Tome 2 (2010) no. 4, pp. 473-496. doi : 10.1142/S1793744210000259. https://cml.centre-mersenne.org/articles/10.1142/S1793744210000259/
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