[Paper Review] Infinitely presented graphical small cancellation groups are acylindrically hyperbolic

Dominik Gruber, Alessandro Sisto|arXiv (Cornell University)|Aug 19, 2014

Geometric and Algebraic Topology39 references17 citations

TL;DR

This paper proves that infinitely presented graphical Gr(7)-small cancellation groups are acylindrically hyperbolic, extending to classical C(7) and C′(1/6)-groups. The authors construct a hyperbolic space via coning-off the Cayley graph and use geometric group theory techniques to establish acylindrical hyperbolicity, yielding new examples of groups with exotic divergence functions and non-relatively hyperbolic structures.

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.

Motivation & Objective

• To establish acylindrical hyperbolicity for infinitely presented graphical Gr(7)-small cancellation groups.
• To extend the result to classical C(7) and C′(1/6)-groups, which are special cases of graphical presentations.
• To construct new examples of groups with complex divergence functions, including superlinear growth exceeding any subexponential function.
• To demonstrate the existence of finitely generated non-relatively hyperbolic groups containing non-degenerate hyperbolically embedded subgroups.
• To provide a constructive method for embedding any finitely generated infinite group as a non-degenerate hyperbolically embedded subgroup in a non-relatively hyperbolic group.

Proposed method

• Define graphical small cancellation groups G(Γ) from labelled graphs Γ, where relators are words read along closed paths.
• Use the coning-off construction on the Cayley graph of G(Γ) to produce a G(Γ)-action on a hyperbolic space Y.
• Apply geometric criteria for acylindrical hyperbolicity by analyzing intersection patterns of embedded cycle graphs in the Cayley graph.
• Construct explicit paths and cycles to show that the action on Y is acylindrical and has unbounded orbits.
• Use relative Dehn functions and diagrammatic arguments to verify hyperbolically embedded subgroups in the constructed groups.
• Leverage small cancellation over free products to generate groups with controlled asymptotic and geometric properties.

Experimental results

Research questions

• RQ1 Are infinitely presented graphical Gr(7)-groups acylindrically hyperbolic?
• RQ2 Can the divergence function of a finitely generated group exhibit superlinear growth while exceeding any subexponential function?
• RQ3 Can every finitely generated infinite group be embedded as a non-degenerate hyperbolically embedded subgroup in a non-relatively hyperbolic group?
• RQ4 Do classical C′(1/6)-groups constructed via graphical small cancellation exhibit acylindrical hyperbolicity?
• RQ5 What are the geometric and asymptotic properties of groups built via small cancellation over free products?

Key findings

• Infinitely presented graphical Gr(7)-labelled groups are acylindrically hyperbolic unless virtually cyclic.
• All infinitely presented classical C(7)-groups and C′(1/6)-groups are acylindrically hyperbolic.
• The paper constructs an explicit example of an infinitely presented classical C′(1/6)-group whose divergence function has limit superior exceeding any given countable set of subexponential functions.
• The same group has a divergence function with limit inferior bounded by a quadratic polynomial.
• The group G(I) in Theorem 5.2 is not hyperbolic relative to any collection of proper subgroups.
• Every finitely generated infinite group embeds as a non-degenerate hyperbolically embedded subgroup in a finitely generated group that is not non-trivially relatively hyperbolic.