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Pure cactus groups and configuration spaces of points on the circle

Takatoshi Hama, Kazuhiro Ichihara

TL;DR

The paper investigates pure cactus groups $PJ_n$ and their ties to configuration spaces on the circle, emphasizing the cases $n=3,4$ and the identification $PJ_n\\cong\\pi_1(\\overline{X(n+1)})$ alongside the Deligne–Mumford real moduli space $\\overline{M_{0,n+1}(\\mathbb{R})}$. It derives explicit presentations for $PJ_4$ via a Dirichlet-polygon construction on the Cayley complex $C_4^{[2,3]}$, leveraging a hyperbolic structure on $\\mathbb{H}^2$ to obtain a complete set of relations through the Poincaré polygon theorem. The authors then prove a direct topological equivalence by constructing a cellular homeomorphism between $C_4^{[2,3]}/PJ_4$ and the combinatorial compactification $\\overline{X(5)}$, showing $PJ_4 \\cong\\pi_1(\\overline{X(5)})$ and detailing the 12 pentagonal 2-cells that encode the space's structure. Finally, the work raises open questions about the hyperbolicity and growth of higher pure cactus groups, notably $PJ_5$ and $J_5$, and whether these properties can be established without recourse to $\\overline{M_{0,n+1}(\\mathbb{R})}$, highlighting connections to real moduli geometry and potential computational challenges.

Abstract

In this article, we provide a summary of the results presented in the previous two papers of the authors and in the second author's Master thesis, which concern pure cactus groups and configuration spaces of points on the circle.

Pure cactus groups and configuration spaces of points on the circle

TL;DR

The paper investigates pure cactus groups and their ties to configuration spaces on the circle, emphasizing the cases and the identification alongside the Deligne–Mumford real moduli space . It derives explicit presentations for via a Dirichlet-polygon construction on the Cayley complex , leveraging a hyperbolic structure on to obtain a complete set of relations through the Poincaré polygon theorem. The authors then prove a direct topological equivalence by constructing a cellular homeomorphism between and the combinatorial compactification , showing and detailing the 12 pentagonal 2-cells that encode the space's structure. Finally, the work raises open questions about the hyperbolicity and growth of higher pure cactus groups, notably and , and whether these properties can be established without recourse to , highlighting connections to real moduli geometry and potential computational challenges.

Abstract

In this article, we provide a summary of the results presented in the previous two papers of the authors and in the second author's Master thesis, which concern pure cactus groups and configuration spaces of points on the circle.
Paper Structure (5 sections, 6 theorems, 30 equations, 8 figures)

This paper contains 5 sections, 6 theorems, 30 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Configuration spaces of points on $S^1$
  3. A presentation of $PJ_4$
  4. $PJ_4$ and $\overline{X(5)}$
  5. Questions

Key Result

Theorem 1

Let $\widetilde{X(4)}$ be the universal cover of the space $\overline{X(4)}$, endowed with the action $\widetilde{\Gamma}$ of the fundamental group $\pi_1 (\overline{X(4)} )$ of $\overline{X(4)}$ as deck transformations. Let ${C_3}^{\{ 2\}}$ be the Cayley complex of the subgroup $J_3^{\{2\}}$ of the

Figures (8)

  • Figure 1: Diagrams for some elements of $J_4$
  • Figure 2: Neighborhood of $e$ in the Cayley complex ${C_4}^{[ 2, 3 ]}$
  • Figure 3: The $\{4,5\}$-tesselation of the hyperbolic plane. (BandeltChepoiEppstein)
  • Figure 4:
  • Figure 5: Five Möbius bands
  • ...and 3 more figures

Theorems & Definitions (7)

  • Theorem 1: HamaIchiharaPJ3
  • Corollary 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • proof : Proof of Theorem \ref{['PJ4 and X5']}