Connected components of definable groups, and o-minimality II

In this sequel to [3] we try to give a comprehensive account of the “connected components” G00 and G000 as well as the various quotients G/G00, G/G000, G00/G000, for G a group definable in a (saturated) o-minimal expansion of a real closed field. Key themes are the structure of G00/G000 and the problem of “exactness” of the G↦G00 functor. We prove that the examples produced in [3] are typical, and that for any G, G00/G000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup (where we allow the trivial Lie group).