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Rational cross-sections, bounded generation and orders on groups

Corentin Bodart

TL;DR

This work investigates the existence of rational cross-sections in finitely generated groups by linking rational cross-sections to bounded generation and left-invariant rational orders. It establishes positive results for wreath products when the acting group $Q$ is virtually $(R+LO)$ and the base $L$ has a rational cross-section, and conversely provides broad negative criteria showing many wreath products and torsion-by-$Z$ groups lack rational cross-sections. The paper constructs explicit non-examples, including extensions of Grigorchuk-type groups and Houghton’s group $H_2$, and proves the first finitely presented group with solvable word problem that has no rational cross-section via a finitely presented extension of Grigorchuk’s group. These results deepen the understanding of the limitations of rational normal forms and have implications for autostackability and rewriting systems in group theory.

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.

Rational cross-sections, bounded generation and orders on groups

TL;DR

This work investigates the existence of rational cross-sections in finitely generated groups by linking rational cross-sections to bounded generation and left-invariant rational orders. It establishes positive results for wreath products when the acting group is virtually and the base has a rational cross-section, and conversely provides broad negative criteria showing many wreath products and torsion-by- groups lack rational cross-sections. The paper constructs explicit non-examples, including extensions of Grigorchuk-type groups and Houghton’s group , and proves the first finitely presented group with solvable word problem that has no rational cross-section via a finitely presented extension of Grigorchuk’s group. These results deepen the understanding of the limitations of rational normal forms and have implications for autostackability and rewriting systems in group theory.

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 and , a group of permutations of , 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.
Paper Structure (18 sections, 36 theorems, 67 equations, 15 figures)

This paper contains 18 sections, 36 theorems, 67 equations, 15 figures.

Table of Contents

  1. Background information
  2. Regular languages
  3. Rational subsets and cross-sections
  4. Left-invariant orders
  5. Intersections of rational subsets
  6. Monsters without rational cross-sections
  7. Extensions of infinite torsion groups
  8. Groups acting on regular rooted trees
  9. Orders and wreath products: Positive results
  10. Orders and wreath products: Negative results
  11. Main lemma
  12. A criterion for wreath products
  13. Rational cones in free and infinite-ended groups
  14. Torsion-by-$\mathbb Z$ groups and Houghton's $H_2$
  15. No rational cross-section for Houghton's group $H_2$
  16. ...and 3 more sections

Key Result

Theorem 1

Consider a product $A*_CB$ with $C$ finite, s.t. either $A$ or $B$ doesn't have a rational cross-section. Then neither does $A*_CB$.

Figures (15)

  • Figure 1: $M$ for $\mathcal{L}=\{\varepsilon\}\cup\{t^n, T^n\mid n\geqslant 1\}$
  • Figure 2: $M$ for $\mathcal{L}=\{t^n\mid n\equiv 1,2\pmod 3\}$
  • Figure 3: $M$ for $\mathcal{L}\underset{\mathop{\mathrm{ev}}\nolimits}{\longleftrightarrow} C_2\wr \mathbb Z$
  • Figure 4: $w$ asymmetrically $K$-fellow travel with $v$. We see $w$ is strongly restrained by $v$ while conditions imposed on $v$ by $w$ are much weaker.
  • Figure 5: Part of $\tilde{M}$ for $G=\left\langle a,b\mid [a,b],b^2\right\rangle$, $\mathcal{A}=\{a,A=a^{-1},b\}$ and $K=1$. Type 1 edges can be seen in black, type 2 in purple, and some type 3 in pink.
  • ...and 10 more figures

Theorems & Definitions (70)

  • Theorem : GerstenShort, see also Theorem \ref{['sec2:nightmare']}
  • Theorem 1: Theorem \ref{['sec4:positive_result']}
  • Theorem 2
  • Theorem 3: Theorem \ref{['sec6:Houghton_is_down']}
  • Theorem 4: Theorem \ref{['sec7:Lysenok_is_down']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: Kleene's theorem
  • Definition 1.6
  • ...and 60 more