Integral criteria of hyperbolicity for graphs and groups

TL;DR

This paper introduces three new criteria based on average widths of geodesic bigons to determine hyperbolicity in graphs and groups, linking geometric properties to algebraic hyperbolic behavior.

Contribution

It provides novel integral criteria for hyperbolicity of graphs and groups, connecting Van Kampen area ratios to hyperbolic classification.

Findings

Bounded ratio of Van Kampen area to geodesic length implies hyperbolicity.

Criteria applicable to Cayley graphs of finitely presented groups.

Potential characterization of hyperbolic groups via random walks.

Abstract

We establish three criteria of hyperbolicity of a graph in terms of ``average width of geodesic bigons''. In particular we prove that if the ratio of the Van Kampen area of a geodesic bigon $\beta$ and the length of $\beta$ in the Cayley graph of a finitely presented group $G$ is bounded above then $G$ is hyperbolic. We plan to use these results to characterize hyperbolic groups in terms of random walks.