The Avalanche Principle and negative curvature

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.