Commutativity in groups presented by finite Church-Rosser Thue systems

Rewriting systems, plain groups, and geodetic graphs

“…Early progress was swift. Diekert [4] (see also [11]) proved that that the family of groups admitting presentation by finite convergent length-reducing rewriting systems is properly contained within the family of virtually-free groups; Avenhaus, Madlener and Otto [1] proved that the family of groups admitting presentation by finite convergent length-reducing rewriting systems in which each rule has a left-hand-side of length two is exactly the family of plain groups (a group is plain if it isomorphic to a free product of finitely-many factors, with each factor a finite group or an infinite cyclic group); an explicit construction (described in Section 2.1) shows that any plain group admits presentation by a finite convergent length-reducing rewriting system. From such results the plain groups emerged as the likely family of groups presented by finite convergent length-reducing rewriting systems.…”