Relative Hyperbolicity of Graphical Small Cancellation Groups

Abstract

A piece of a labelled graph Γ is a labelled path that embeds into Γ in two essentially different ways. We prove that graphical Gr ( 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 Γ, if and only if the pieces of Γ are uniformly bounded. This implies the relative hyperbolicity by a result of C. Drut ¸u, D. Osin and M. Sapir.