Relative Hyperbolicity of Graphical Small Cancellation Groups

Abstract: 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. Dru\c{t}u, D. Osin and M. Sapir.