Infinitely presented C(6)-groups are SQ-universal

TL;DR

This paper proves that certain infinitely presented small cancellation groups are SQ-universal and constructs many non-quasi-isometric groups with specific presentation properties.

Contribution

It extends SQ-universality results to graphical small cancellation groups and constructs uncountably many distinct groups with particular presentation constraints.

Findings

Infinitely presented $C(6)$ groups are SQ-universal.

Extension to graphical $Gr_*(6)$-groups over free products.

Existence of uncountably many non-quasi-isometric groups with $C(p)$-presentations but no graphical $Gr'(rac{1}{6})$-presentations.

Abstract

We prove that infinitely presented classical $C(6)$ small cancellation groups are SQ-universal. We extend the result to graphical $Gr_*(6)$-groups over free products. For every $p\in\mathbb{N}$, we construct uncountably many pairwise non-quasi-isometric groups that admit classical $C(p)$-presentations but no graphical $Gr'(\frac{1}{6})$-presentations.