# The Avalanche Principle and negative curvature

## TL;DR

This paper offers a new geometric proof of the Avalanche Principle using hyperbolic geometry, extending its applicability to general CAT(-1) spaces and deriving a polygonal Schur theorem.

## Contribution

It introduces a geometric approach to the Avalanche Principle, broadening its scope to CAT(-1) spaces and establishing a related polygonal Schur theorem.

## Key findings

- New proof of the Avalanche Principle using hyperbolic geometry
- Extension of the principle to CAT(-1) metric spaces
- Derivation of a polygonal Schur theorem for these spaces

## Abstract

We use the geometric structure of the hyperbolic upper half plane to provide a new proof of the Avalanche Principle introduced by M. Goldstein and W. Schlag in the context of $\mathrm{SL}_{2}(\mathbb{R})$ matrices. This approach allows to interpret and extend this result to arbitrary $\mathrm{CAT}(-1)$ metric spaces. Through the proof, we deduce a polygonal Schur theorem for these spaces.