Ilie Ugarcovici (Penn State)

On admissible geometric codes for geodesics on modular surfaces

Abstract. Two different methods are available for coding geodesics on surfaces of constant negative curvature: a geometric one (Morse code) obtained by keeping track of the sides of a fundamental region hit by the geodesic, and an arithmetic one (Artin code) obtained by coding the endpoints of the geodesic (using continued fractions).

In this talk, we give a sufficient condition for a finite sequence of integers to be realizable as the geometric code of a closed geodesic on the modular surface. We will also discuss the problem for other modular surfaces, in particular for $\mathcal{H}/\Gamma(2)$, where $\mathcal H$ is the hyperbolic plane and $\Gamma(2)$ is the principal congruence subgroup of level 2.

This is joint work with Svetlana Katok.