I was wondering about the following question: if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct a new action such that a point p in R has the trivial stabilizer in G. Is it possible to make (possibly by a completely new action) the map G -> R defined by g -> g(p) is a group homomorphism always (when R is regareded as a group with addition)? If not, when could it be done?
Any comment and/or advice would be greatly appreciated.
Edit: As many people pointed out already, if such a homomorphism G -> R exists, then G must be abelian - I was over-simplifying the question. What I really want is when G is a countable group acting faithfully on R and S is a finite subset of G, can I construct a new action so that a point p with trivial stabilzer satisfies that g(p) + h(p) = g(h(p)) for all g, h in S?