Beyond Expansion II: Low-Lying Fundamental Geodesics

TL;DR

This paper proves the existence of infinitely many low-lying fundamental geodesics on the modular surface, connecting geometric properties with algebraic class group elements, and answers a significant open question in the field.

Contribution

It establishes the infinite existence of low-lying fundamental geodesics, linking geometric and algebraic structures in number theory.

Findings

Infinitely many low-lying fundamental geodesics exist.

Answers a question posed by Einsiedler-Lindenstrauss-Michel-Venkatesh.

Bridges geometric and algebraic aspects of quadratic fields.

Abstract

A closed geodesic on the modular surface is "low-lying" if it does not travel "high" into the cusp. It is "fundamental" if it corresponds to an element in the class group of a real quadratic field. We prove the existence of infinitely many low-lying fundamental geodesics, answering a question of Einsiedler-Lindenstrauss-Michel-Venkatesh.