# Qualitative counting closed geodesics

## TL;DR

This paper explores the geometric properties of word metrics on fundamental groups of manifolds, focusing on the abundance or scarcity of closed geodesics and their impact on the metric's diameter.

## Contribution

It introduces a framework to analyze the finiteness or infiniteness of the diameter of word metrics based on closed geodesic properties.

## Key findings

- Finite diameter indicates abundance of closed geodesics
- Infinite diameter indicates scarcity of closed geodesics
- Provides examples illustrating both cases

## Abstract

We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the second one as their scarcity. We discuss examples for both cases.