Rewriting systems, plain groups, and geodetic graphs

Abstract

We take a new step towards an algebraic characterisation of groups presented by length-reducing rewriting systems. We prove that a group is presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length three if and only if the group is plain.Our proof rests on proving a new result about embedded circuits in geodetic graphs, whose proof may also be of independent interest to graph theorists.