# An arithmetic property of the set of angles between closed geodesics on hyperbolic surfaces of finite type

## TL;DR

This paper proves that on finite-type hyperbolic surfaces, the set of angles between closed geodesics contains only finitely many rational multiples of pi, revealing a fundamental geometric property.

## Contribution

It establishes a finiteness result for rational multiples of pi in the set of angles between closed geodesics on hyperbolic surfaces of finite type.

## Key findings

- Only finitely many rational multiples of pi are in the set of angles.
- The set of angles between closed geodesics is algebraically constrained.
- The result deepens understanding of geometric structures on hyperbolic surfaces.

## Abstract

For a hyperbolic surface S of finite type we consider the set A(S) of angles between closed geodesics on S. Our main result is that there are only finitely many rational multiples of \pi in A(S).