Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

TL;DR

This paper introduces a new metric condition ${\LARGE{\tau}}'$ that generalizes small cancellation theory, applies to various groups including Artin groups, and establishes properties like diagrammatic reducibility, asphericity, quadratic Dehn functions, and hyperbolicity.

Contribution

It defines the ${\LARGE{\tau}}'$ condition, proves its implications for group properties, and explores metric, non-metric, and dual variants, extending small cancellation theory.

Findings

Groups satisfying ${\LARGE{\tau}}'$ are diagrammatically reducible.

Under certain conditions, ${\LARGE{\tau}}'$-groups have quadratic Dehn functions.

The ${\LARGE{\tau}}'_{<}$ condition implies hyperbolicity.

Abstract

We present a metric condition ${\LARGE{\tau}}'$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition ${\LARGE{\tau}}'$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, ${\LARGE{\tau}}'$-groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation $P=\langle X \mid R\rangle$ of group $G$ satisfies conditions ${\LARGE{\tau}}'-C'(\frac{1}{2})$, the length of any nontrivial word in the free group generated by $X$ representing the trivial element in $G$ is at least that of the shortest relator. We also introduce a strict metric condition ${\LARGE{\tau}}'_{<}$, which implies hyperbolicity. Finally we investigate non-metric and dual variants of these conditions, and study a harmonic mean version of small cancellation theory.