@YvesCornulier Nice one. So the answer is no for general cases. Thanks!

@YvesCornulier Thanks. Then, let's say that $X$ is a compact metric space. (the terms limit point / accumulation point / cluster point messed up ... )

@Humphreys: This happens to be one of the papers that leads us to our current focus. Thanks.

Thank you very much for your answer (and also for your book of course :P). A'Campo's note is among the documents on my desk. Seems that I didn't miss anything. Then I think I'll start to look into it. For your information, the Coxeter groups that I'm particularly interested in are those of type (n-1,1) ("lorentzian" or "hyperbolic" in literature). We may have observed something interesting about their eigenvectors, but nothing is clear yet.

@David Cohen: Yes of course. But we need to assume that an embedding exists at the first place. I mean, situation like $G=C_3$ (triangle), $\ell=3,4,10$ should not be considered.

any one tell me how to type the braket {} for a set? \{ \} doesn't work.