Arithmeticity in hyperbolic geometry

• Grant number: 24/01650-3
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: April 30, 2024
• End date: April 29, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: André Salles de Carvalho
• Grantee: Gregory Cosac Daher
• Supervisor: Alan William Reid
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Institution abroad: Rice University, United States
• Associated to the scholarship: 22/10772-0 - Arithmeticity in low dimensional geometry, BP.PD

Abstract

The focus of my research is to study arithmetic properties of Fuchsian groups in order to better understand geometric invariants of their associated hyperbolic surfaces. This project is built around the question of determining the commensurability class of a Fuchsian group from its length spectrum. More precisely, we aim to apply the techniques developed in the work of A. Reid to a broader class of Fuchsian groups. This project is also concerned with the characterisation of arithmetic and semi-arithmetic Fuchsian groups through the growth rate of their trace set, as proposed in the works of Luo, Sarnak and Schmutz Schaller, and further developed by Geninska and Leuzinger.