Closed geodesics and bounded gaps

TL;DR

This paper explores the bounded gaps property of prime numbers in the context of closed geodesic lengths, showing it holds for some cases but not universally across all Teichmüller space.

Contribution

It establishes the bounded gaps property for closed geodesic lengths in certain cases and demonstrates its failure in a dense subset of Teichmüller space.

Findings

Bounded gaps property holds for congruence subgroups.

The property fails for a dense set in Teichmüller space.

Geometric analogue of prime numbers via Selberg zeta functions.

Abstract

The bounded gaps property of the prime numbers, as proven by Yitang Zhang, is considered for sequences of lengths of closed geodesics, which by the theory of Selberg zeta functions are the geometric analogue of the prime numbers. It turns out that the property holds for congruence subgroups and is false for a dense set in Teichm\"uller space.