One-dimensional groups definable in o-minimal structures (Q1339385)

The setting is an o-minimal structure whose domain has definably the structure of an ordered field \(R\) (which will have to be real closed), and there is a definable one-dimensional group \(G\). It is shown that \(G\) is definably isomorphic to an Abelian group \(H\) on (0,1) or \(S'\), with continuous group action. In particular, \(G\) cannot have Euler characteristic 1. It is also shown that if \(R= \mathbb{R}\), then there is a (possibly not definable) group homomorphism \(\phi: (\mathbb{R},+)\to H\) such that if \(H= (0,1)\), then \(\phi\) is a homeomorphism, and if \(H= S'\), then \(\phi\) is a covering.