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Added:
These results continue to hold when $G$ is not compact, see \cite{Illman00}.
How does comparison with the plain case, triangulation theorem, look with regard to:
Conversely, deep theorems assert that a given kind of triangulation does not generally exists for a given class of manifolds.
My understanding is that the equivariant case is much harder and that there is not essentially more work on it than listed in the entry (good though that Dmitri found one more reference previously missing), which all focuses on (equivariant) smooth triangulations of smooth manifolds.
Of course, a trivial thing to be said is that when a given type of ordinary triangulation does not exist, then an equivariant version won’t exist either.
There are some remarks at MO here.
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