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Preprint Number 243

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243. Davide Penazzi
One-basedness and groups of the form G/G^{00}

Submission date: 27 April 2010.


We initiate a geometric stability study of groups of the form G/G^{00}, where G is a 1-dimensional definably compact, definably connected, definable group in a real closed field M. We consider an enriched structure M' with a predicate for G^{00} and prove 1-basedness for additive truncations of M, multiplicative truncations, SO_2(M) and its truncations; such groups are now interpretable in M'. We prove that the only 1-based groups are sufficiently “big” multiplicative truncations, and we relate the results obtained to valuation theory. In the last section we extend our results to ind-definable groups constructed from these above.

Mathematics Subject Classification: 03C45, 03C64, 14P99

Keywords and phrases: G^{00}, bounded hyperdefinable groups, 1-basedness, real closed valued fields

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