# Stable Strata of Geodesics in Outer Space

## TL;DR

This paper introduces an Outer space analogue of the principal stratum of Teichmüller space, analyzing the stability and stratification of geodesics, revealing both similarities and new pathological phenomena.

## Contribution

It defines a new stratification in Outer space analogous to the principal stratum in Teichmüller space and studies its stability and unique pathological features.

## Key findings

- Outer space geodesics share stability properties with Teichmüller space geodesics.
- The stratification of periodic geodesics in Outer space shows new pathological phenomena.
- Some properties of geodesic stratification differ fundamentally from the Teichmüller space case.

## Abstract

In this paper we propose an Outer space analogue for the principal stratum of the unit tangent bundle to the Teichm\"uller space $\mathcal{T}(S)$ of a closed hyperbolic surface $S$. More specifically, we focus on properties of the geodesics in Teichm\"uller space determined by the principal stratum. We show that the analogous Outer space "principal" periodic geodesics share certain stability properties with the principal stratum geodesics of Teichm\"uller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichm\"uller space context.