Weight tests and hyperbolic groups

Summary

Abstract

The notion of reduced diagram plays a fundamental role in small cancellation theory and in tests for detecting the asphericity of 2-complexes. By introducing vertex reduced as a stricter form of reducedness in diagrams we obtain a new combinatorial notion of asphericity for 2-complexes, called vertex asphericity, which generalizes diagrammatic reducibility and implies diagrammatic asphericity. This leads to a generalization and simplification in applying the weight test [2] and the cycle test [6] [7] to detect asphericity of 2-complexes and (for the hyperbolic versions of these tests) to detect hyperbolic group presentations. In the end, we present an application to labeled oriented graphs. We would like to thank the referee for his helpful suggestions.

Basic Definitions

A p.l. map between 2-complexes is called combinatorial, if each open cell is mapped homeomorphically onto its image. A 2-dimensional finite CW-complex is called combinatorial, if the attaching maps of the 2-cells are combinatorial relative to a suitable polygonal subdivision of their boundary.

Let K_P be the standard 2-complex of the presentation P (we assume all presentations to be finite). A diagram is a combinatorial map f: M → K_P, where M is a combinatorial subcomplex of an orientable 2-manifold. A spherical diagram is a diagram f:S → K_P, where S is the 2-sphere. These definitions may be found for example in [1], [2], [6], [7] or [8].