I am reading a paper which says has a sentence 'If G is metrizable, with right-invariant compatible metric d_r...'
So I am trying to gauge how general the subsequent statements are, for which it would be useful to know what kinds of groups admit a right invariant compatible metric. Perhaps one can be constructed for all?
It's a classic theorem (due to Birkhoff, IIRC) that a metrisable topological group $G$ (easily recognisable, because this is eqivalent to having a countable local base at $e$, plus being $T_0$ or better) has a right-invariant metric, compatible with the topology, and also a , possibly different (!), left-invariant metric. Some groups that are metrisable do not have a two-sided invariant metric IIRC. More advanced books in topological group theory will give the construction (and sometimes an example of the last as well). For non-$T_0$ we get invariant pseudometrics instead (but this isn't studied much).