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.

  • $\begingroup$ Is it possible to improve the example such that $\rho$ is injective? $\endgroup$ Sep 23 '14 at 20:14
  • $\begingroup$ I suspect so, but I don't know of an example immediately. $\endgroup$
    – Ian Agol
    Sep 24 '14 at 3:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.