Relative Hyperbolicity of Graphical Small Cancellation Groups

TL;DR

This paper proves that certain graphical small cancellation groups are relatively hyperbolic when their defining graph's pieces are uniformly bounded, using asymptotic tree-gradedness of their Cayley graphs.

Contribution

It establishes the relative hyperbolicity of graphical small cancellation groups under specific boundedness conditions on pieces, extending understanding of their geometric properties.

Findings

Groups are relatively hyperbolic if pieces are uniformly bounded.

Cayley graphs are asymptotically tree-graded with respect to embedded components.

The result connects small cancellation conditions with relative hyperbolicity.

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.