Parabolic and Non-Parabolic Surfaces with Small or Large End Spaces via Fenchel-Nielsen Parameters

Author

Michael Antony Pandazis, The Graduate Center, City University of New YorkFollow

Date of Degree

6-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Dragomir Saric

Committee Members

Ara Basmajian

Enrique Pujals

Subject Categories

Geometry and Topology

Keywords

parabolicity, surfaces, geometry

Abstract

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface X that determine whether or not a surface X is parabolic. Fix a geodesic pants decomposition of a surface and call the boundary geodesics in the decomposition cuffs. For a zero or half-twist flute surface, we prove that parabolicity is equivalent to the surface having a covering group of the first kind. Using that result, we give necessary and sufficient conditions on the Fenchel-Nielsen parameters of a half-twist flute surface X with increasing cuff lengths such that X is parabolic. As an application, we determine whether or not each half-twist flute surface in the Hakobyan slice is parabolic. Let X be a Cantor tree surface or a blooming Cantor tree surface. Basmajian, Hakobyan, and Saric proved that if the lengths of cuffs are rapidly converging to zero, then X is parabolic. More recently, Saric proved a slightly slower convergence of lengths of cuffs to zero implies X is not parabolic. We interpolate between the two rates of convergence of the cuffs to zero and find that these surfaces are not parabolic, thus completing the picture.

Recommended Citation

Pandazis, Michael Antony, "Parabolic and Non-Parabolic Surfaces with Small or Large End Spaces via Fenchel-Nielsen Parameters" (2024). CUNY Academic Works.
https://academicworks.cuny.edu/gc_etds/5744