Infinitely presented graphical small cancellation groups are acylindrically hyperbolic

TL;DR

This paper proves that a broad class of infinitely presented small cancellation groups, including classical and graphical variants, are acylindrically hyperbolic, offering new insights into their geometric properties and divergence functions.

Contribution

It establishes acylindrical hyperbolicity for infinitely presented graphical small cancellation groups, extending known results to new classes and providing novel examples of divergence functions.

Findings

Infinitely presented graphical $Gr(7)$ groups are acylindrically hyperbolic.

Classical $C(7)$-groups and $C'(rac{1}{6})$-groups are acylindrically hyperbolic.

Constructs new examples of groups with diverse divergence functions.

Abstract

We prove that infinitely presented graphical $Gr(7)$ small cancellation groups are acylindrically hyperbolic. In particular, infinitely presented classical $C(7)$-groups and, hence, classical $C'(\frac{1}{6})$-groups are acylindrically hyperbolic. We also prove the analogous statements for the larger class of graphical small cancellation presentations over free products. We construct infinitely presented classical $C'(\frac{1}{6})$-groups that provide new examples of divergence functions of groups.