Relative Hyperbolicity of Graphical Small Cancellation Groups

A piece of a labelled graph \(\Gamma\) defined by D. Gruber is a labelled path that embeds into \(\Gamma\) in two essentially different ways. We prove that graphical \(Gr'(\frac{1}{6})\) small cancellation groups whose associated pieces have uniformly bounded length are relative hyperbolic. In fact, we show that the Cayley graph of such group presentation is asymptotically tree-graded with respect to the collection of all embedded components of the defining graph \(\Gamma\), if and only if the pieces of \(\Gamma\) are uniformly bounded. This implies the relative hyperbolicity by a result of C. Druţu, D. Osin and M. Sapir.