Rational cross‐sections, bounded generation, and orders on groups

Abstract

AbstractWe provide new examples of groups without rational cross‐sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Our examples contain a finitely presented HNN‐extension of the first Grigorchuk group. This last group is the first example of finitely presented group with solvable word problem and without rational cross‐sections. It is also not autostackable, and has no left‐regular complete rewriting system.