Integral criteria of hyperbolicity for graphs and groups (Q6656107)

A graph can be defined as a specific path-metric space whose metric is either continuous or discrete, with the values in the set \(\mathbb{N}\) of positive integers. In particular, \textit{P. Papasoglu}, in [Invent. Math. 121, No. 2, 323--334 (1995; Zbl 0834.20040)], proved (among other things) that the hyperbolicity of a connected graph follows from the uniform thinness of the geodesic bigons.\N\NIn the paper under review, the authors, using the ideas of [loc. cit.] and ideas related to the random walks on groups, establish a new criterion of hyperbolicity of a graph. In particular they 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.