[Paper Review] Small cancellation theory and Burnside problem

Rémi Coulon|arXiv (Cornell University)|Feb 27, 2013

Geometric and Algebraic Topology21 references19 citations

TL;DR

This paper presents a geometric approach to small cancellation theory in hyperbolic groups, using quasi-convex subsets and quasi-geodesics to construct an infinite sequence of hyperbolic quotients. By iteratively applying a geometric small cancellation condition, the authors prove that the free Burnside group $\burn rn$ is infinite for sufficiently large odd exponents $n$, offering a new proof of the Novikov-Adian result via hyperbolic geometry and group actions on hyperbolic spaces.

ABSTRACT

In these notes we detail the geometrical approach of small cancellation theory used by T. Delzant and M. Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

Motivation & Objective

• To provide a geometric reinterpretation of small cancellation theory in the context of hyperbolic groups.
• To establish a new proof of the infiniteness of free Burnside groups $\burn rn$ for large odd $n$ using iterative group quotients.
• To extend the method to show that periodic quotients of non-virtually cyclic torsion-free hyperbolic groups are infinite.
• To develop a framework for controlling overlaps between quasi-convex subsets using geometric invariants like Gromov products and diameter bounds.
• To demonstrate that the limit of an ascending sequence of hyperbolic groups remains infinite by preserving hyperbolicity and non-virtual cyclicity through each step.

Proposed method

• Use of $\delta$-hyperbolic spaces and group actions with proper, cocompact isometric actions to define geometric small cancellation.
• Definition of a family $\mathcal{Q}$ of pairs $(H,Y)$, where $Y$ is a $2\delta$-quasi-convex subset and $H$ is a subgroup acting cocompactly on $Y$.
• Introduction of the maximal overlap $\Delta(\mathcal{Q})$ and minimal length $\ell(\mathcal{Q})$ to quantify cancellation conditions.
• Application of stability of quasi-geodesics: any $L$-local $(1,l)$-quasi-geodesic is $l+8\delta$-quasi-convex for sufficiently large $L$.
• Use of Gromov products to bound distances to paths, ensuring that points on a quasi-geodesic remain close to a reference path under controlled overlap.
• Construction of a sequence of groups $G_k$ via quotients by normal subgroups generated by $H$-subgroups, preserving hyperbolicity and non-virtual cyclicity at each step.

Experimental results

Research questions

• RQ1 Can small cancellation theory be rephrased geometrically in the context of hyperbolic groups to prove infiniteness of Burnside groups?
• RQ2 Under what conditions does iterating small cancellation over a hyperbolic group preserve hyperbolicity and non-virtual cyclicity?
• RQ3 How can geometric invariants like Gromov products and quasi-convexity be used to control overlaps between group elements in a quotient construction?
• RQ4 What is the minimal overlap condition (in terms of $\Delta(\mathcal{Q})$ and $\ell(\mathcal{Q})$) that ensures the resulting quotient remains infinite?
• RQ5 Can the method be extended to show that periodic quotients of arbitrary non-virtually cyclic hyperbolic groups are infinite?

Key findings

• The free Burnside group $\burn rn$ is infinite for all odd $n \geq n_0$, where $n_0$ depends on the group and the small cancellation parameters.
• The construction yields a sequence of groups $G_k$ that are non-virtually cyclic and hyperbolic, ensuring the limit group cannot be finite.
• The geometric small cancellation condition $C''(\lambda)$ is reformulated using quasi-convex subsets and Gromov products, with $\Delta(\mathcal{Q}) \leq \lambda \ell(\mathcal{Q})$ as the key criterion.
• Stability of quasi-geodesics ensures that any $L$-local $(1,l)$-quasi-geodesic is uniformly close to a global quasi-geodesic, with bounds depending only on $l$ and $\delta$.
• The limit group $\burn rn$ is shown to be infinite because it is the direct limit of an ascending sequence of non-virtually cyclic hyperbolic groups.
• The method avoids reliance on metric actions of $\burn rn$ by instead analyzing the direct limit of a geometrically controlled quotient sequence.