Rational cross-sections, bounded generation and orders on groups

TL;DR

This paper constructs new examples of groups lacking rational cross-sections by exploring connections with bounded generation and rational orders, including a finitely presented group with unique properties.

Contribution

It introduces the first finitely presented group with solvable word problem that lacks rational cross-sections, expanding understanding of group structures.

Findings

Examples of groups without rational cross-sections are provided.

A finitely presented group with solvable word problem and no rational cross-sections is constructed.

The group is not autostackable and lacks a left-regular complete rewriting system.

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.