On a generalization of Dehn's algorithm

TL;DR

This paper generalizes Dehn's algorithm as a rewriting system to broader classes of groups, including nilpotent and relatively hyperbolic groups, and explores conditions under which such algorithms cannot exist.

Contribution

It introduces Cannon's algorithms, extending Dehn's algorithm to new group classes and analyzes their closure properties and limitations.

Findings

Includes nilpotent and relatively hyperbolic groups

Shows closure properties of the generalized algorithm class

Identifies conditions preventing the existence of such algorithms

Abstract

Viewing Dehn's algorithm as a rewriting system, we generalise to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.