One-basedness and groups of the form G/G00

Abstract

We initiate a geometric stability study of groups of the form G/G00, 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 G00 and check 1-basedness or non-1-basedness for G/G00, where G is an additive truncation of M, a multiplicative truncation of M, SO2(M) or one of its truncations; such groups G/G00 are now interpretable in M′. We prove that the only 1-based groups are those where G is a sufficiently “big” multiplicative truncation, and we relate the results obtained to valuation theory. In the last section we extend our results to ind-hyperdefinable groups constructed from those above.