# On the realization of a compact surface as a leaf of an analytic foliation

Let $S$ be a compact orientable surface of genus $g \geq 2$. Is there any transversely real (or complex) analytic codimension one foliation $\mathcal{F}$ such that $\mathcal{F}$ has $S$ as a leaf with an infinite holonomy group?

Sure, take a homomorphism of $\rho:\pi_1(S)\to Diff^\omega(\mathbb{R})$ which has a global fixed point (for example, it might factor through $\mathbb{Z}$). Take the diagonal quotient $\mathbb{H}^2\times \mathbb{R}$, where $\pi_1(S)$ acts diagonally (on $\mathbb{H}^2$ as a fuchsian group, and on $\mathbb{R}$ by $\rho$). Then the quotient will be an analytic manifold with a closed surface leaf with non-trivial holonomy.
• Is it possible to improve the example such that $\rho$ is injective? Sep 23 '14 at 20:14