Rewriting systems, plain groups, and geodetic graphs

TL;DR

This paper characterizes plain groups as those presented by finite convergent rewriting systems with rules of length three, using novel insights into geodetic graph circuits.

Contribution

It establishes a new equivalence between plain groups and specific rewriting systems, introducing a graph-theoretic approach involving geodetic graphs.

Findings

Plain groups are characterized by finite convergent rewriting systems with length-3 rules.

A new property of embedded circuits in geodetic graphs is proven.

The results connect group theory with graph theory in a novel way.

Abstract

We prove that a group is presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 if and only if the group is plain. Our proof goes via a new result concerning properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.