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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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