Real Algebraic and Analytic Geometry |

Lie-like decompositions of groups definable in o-minimal structures.

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Submission: 2009, December 23.

*Abstract:
There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups.
We study definable groups G which are not definably compact showing that they
have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H
such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact.
In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.*

Mathematics Subject Classification (2000): 03C64, 22E15.

Keywords and Phrases: o-minimality, definable groups, real Lie groups.

**Full text**, 44p.:
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pdf 849k.

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