### Session S39 - Differential Equations and Geometric Structures

Tuesday, July 13, 12:00 ~ 12:50 UTC-3

## A family of parabolic tight flute surfaces

### Camilo Ramírez Maluendas

#### Universidad Nacional de Colombia, Sede Manizales, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

A Riemann surface $S$ is called parabolic if and only if there are no nonconstant negative subharmonic functions on $S$. From the Uniformization Theorem any Riemann surface can be thought as the quotient space of the hyperbolic plane $\mathbb{H}$ by a Fuchsian group $\Gamma$, i.e. $S=\mathbb{H}/\Gamma$. Such surfaces admit a Riemannian metric of constant curvature. So, the Riemman surface $S$ is parabolic if and only if it satisfies one of the following conditions:

1. The behaviour of the geodesic flow on the unit tangent bundle of $S$ is ergodic;

2. The Poincaré series of $\Gamma$ diverges;

3. The Fuchsian group $\Gamma$ has the Mostow rigidity property;

4. The Riemann surface $S$ has the Bowen's property;

5. Almost every geodesic ray of $S$ is recurrent.

In a recent work, A. Basmajian, H. Hakobyan and D. Šaric´ have described sufficient condition to determinate the parabolicity of certain Riemann surface of infinite-type from the Fenchel-Nielsen conditions.

In this talk, we will use the Fenchel-Nielsen conditions and build geometrically a family (moduli space) conformed by tight flute surfaces (which are topologically equivalent to the infinite-type genus zero surface whose ends space is homeomorphic to the closure $\overline{\{1/n:n\in\mathbb{N}\}}$), such that each one of these is parabolic.

References

[1] J. A. Arredondo and C. Ramírez Maluendas. On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree. Comp. Man. 7 (2020), no. 1, 73-92.

[2] L. Ahlfors and L. Sario. Riemann surfaces. Princeton Mathematical Series, No. 26 Princeton University Press, Princeton, N.J. 1960.

[3] A. Basmajian. Hyperbolic structures for surfaces of in infinite type. Trans. Amer. Math. Soc. 336, no. 1, March 1993, 421-444.

[4] A. Basmajian, H. Hakobyan and D. Šaric´. The type problem from Riemann surfaces via Fenchel-Nielsen parameters. Preprint, available on arXiv.

Joint work with John Alexander Arredondo García (Fundación Universitaria Konrad Lorenz, Bogotá, Colombia) and Israel Morales Jiménez (Instituto de Matemáticas UNAM, Unidad Oaxaca, Oaxaca, México).