Rational cross-sections, bounded generation and orders on groups

Abstract

We provide new examples of groups without rational cross-sections (also called regular normal forms), using connections with bounded generation and rational orders on groups. Specifically, our examples are extensions of infinite torsion groups, groups of Grigorchuk type, wreath products similar to $C_2\wr (C_2\wr \mathbb{Z})$ and $\mathbb{Z}\wr F_2$, a group of permutations of $\mathbb{Z}$, and 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.

Fixed a mistake in the proof of Proposition 7.4, and other minors changes

References in corpus (3)

• Hausdorff dimension of some groups acting on the binary tree
• Geometry of the word problem for 3-manifold groups
• The Relaxation Normal Form of Braids is Regular