  Let $D$ be a domain in $C^n$ and $G$ a topological group which acts
effectively on $D$ by holomorphic automorphisms. In this paper we are
interested in projective linearizations of the action of $G$, i.e. a linear
representation of $G$ in some $C^{N+1}$ and an equivariant imbedding of $D$
into $\P^N$ with respect to this representation. The domains we discuss here
are open connected sets defined by finitely many real polynomial inequalities
or connected finite unions of such sets. Assume that the group $G$ acts by
birational automorphisms.
  Our main result is the equivalence of the following conditions:
  1) there exists a projective linearization, i.e. a linear representation of
$G$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow
\P^N$ such that the restriction $i|_D$ is $G$-equivariant.
  2) $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of
$D$ which extends the action of $G$ and has finitely many connected components;
  3) $G$ is a subgroup of a Nash group $\hat G$ of birational automorphisms of
$D$ which extends the action of $G$ to a Nash action $\hat G\times D\to D$;
  4) $G$ is a subgroup of a Nash group $\hat G$ such that the action $G\times
D\to D$ extends to a Nash action $\hat G\times D\to D$;
  5) the degree of the automorphism $\phi_g\colon D\to D$
