# Structural stability of meandering-hyperbolic group actions

## TL;DR

This paper introduces a new concept called meandering hyperbolicity for group actions on metric spaces, extending Sullivan's stability results to a broader class including some non-hyperbolic groups.

## Contribution

It relaxes Sullivan's axioms to define meandering hyperbolicity and proves that such actions are structurally stable, broadening the scope of stability results.

## Key findings

- Meandering-hyperbolic actions are structurally stable.
- The notion includes actions of certain non-hyperbolic groups.
- Basic properties and examples of meandering-hyperbolic actions are provided.

## Abstract

In his 1985 paper Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper we relax Sullivan's axioms and introduce a notion of "meandering hyperbolicity" for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain non-hyperbolic groups, such as actions of "uniform lattices" in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.