Large genus bounds for the distribution of triangulated surfaces in moduli space

Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the genus tends to infinity. Mirzakhani raised this question in her 2010 ICM address. We show that in the large genus case, triangulated surfaces are well distributed in moduli space in a fairly strong sense. We do this by proving upper and lower bounds for the number of triangulated surfaces lying in a Teichmüller ball in moduli space. In particular, we show that the number of triangulated surfaces lying in a Teichmüller unit ball is at most exponential in the number of triangles, independent of the genus.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2022

Author dc:creator

• Vasudevan, Sahana

Advisor dc:contributor.advisor

• Guth, Larry

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright MIT

Licence dc:rights.uri

• http://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/145025

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/145025