One-basedness and groups of the form G/G 00

We initiate a geometric stability study of groups of the form G/G, 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 and prove 1-basedness for additive truncations of M , multiplicative truncations, SO2(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.