I am reading Ghys' paper Holomorphic Anosov System. In the beginning of section 5, there is the following argument:
Let $\phi$ be a holomorphic Anosov diffeomorphism of a compact manifold $M$. Assume that the unstable foliation $F^u$ has complex dimension 1. We have shown that the stable foliation admits a transversely projective structure. This implies that the lifted foliation $\tilde F^s$ in the universal covering space is defined by a global submersion: $$D:\tilde M\to \mathbb C \mathbb P^1.$$ Moreover,there exists a (holonomy) homomorphism $$H:\pi_1(M)\to PSL(2,\mathbb C)$$ such that for every $\tilde x \in \tilde M$ and $\gamma \in\pi_1(M)$ one has the following property of developing map: $$D(\gamma \cdot \tilde x) =H(\gamma)(D(\tilde x)).$$
As far as I know, in the definition of $(G,X)$-manifold, the target manifold $M$ should have the same dimension with $X$. However, in this case, $M$ has a higher dimension than $\mathbb C\mathbb P^1$, and I don't understand how to define $D$.