Lower bound on the number of non-simple closed geodesics on surfaces

TL;DR

This paper establishes a lower bound on the number of non-simple closed geodesics on hyperbolic surfaces based on length and self-intersection constraints, revealing a transition from polynomial to exponential growth as intersection complexity increases.

Contribution

It provides a novel lower bound for non-simple closed geodesics on hyperbolic surfaces, including explicit constructions on pairs of pants, and analyzes the growth transition related to self-intersection number.

Findings

Lower bound on geodesic count for given length and intersection

Construction method for geodesics on pairs of pants

Transition from polynomial to exponential growth in geodesic number

Abstract

We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for arbitrary surfaces follows from the lower bound on pairs of pants. This lower bound demonstrates that as the self-intersection number $K = K(L)$ goes from a constant to a quadratic function of $L$, the number of closed geodesics transitions from polynomial to exponential in $L$. We show upper bounds on the number of such geodesics in a subsequent paper.