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Preprint Number 413
413. Jakub Gismatullin and Krzysztof Krupinski
On model-theoretic connected components in some group extensions
Submission date: 26 January 2012.
We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We find a certain condition on 2-cocycles guaranteeing that the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some, rather general, situations, our condition also turns out to be necessary for the two connected components to be different.
Using our general results together with the Matsumoto-Moore theory, we obtain new classes of examples of groups whose smallest type-definable subgroup differs from the smallest invariant subgroup. This includes the first known example of a group with this property found by Conversano and Pillay (namely the universal cover of SL_2(R)).
Mathematics Subject Classification: Primary 03C60, Secondary 20A15
Keywords and phrases: model-theoretic connected components, the smallest type-definable subgroup of bounded index, the smallest invariant subgroup of bounded index, G-compactness, universal central extensions, symplectic Steinberg symbols
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