Round twin groups on few strands

TL;DR

This work analyzes ${Q}_n$, the configuration space of $n$ ordered points on the circle with no triple coincidences and a fixed last point, by combining discriminant-complement, cubical-CAT(0), and no-3-equal decompositions. It proves ${Q}_n$ is a $K(\Pi_n,1)$ with $\Pi_n$ the round twin group, computes the first Betti number and Euler characteristic for general $n$, and gives complete Betti-number data for small $n$ (including explicit models for ${Q}_5$, ${Q}_6$, and ${Q}_7$). The paper also connects ${Q}_n$ to moduli spaces ${\overline{\mathcal M}}_{0,n}(\mathbb R)$, embeds $\Pi_n$ into pure cactus groups, and introduces the full round twin group $\Upsilon_{n-1}$ with a concrete presentation, showing injectivity into the full cactus group for $n\ge 4$ and hence residual nilpotence of $\Pi_n$. Collectively these results illuminate the algebraic and topological structure of round twin groups and their relationship to moduli spaces and cactus groups.

Abstract

We study the space $Q_n$ of all configurations of $n$ ordered points on the circle such that no three points coincide, and in which one of the points (say, the last one) is fixed. We compute its fundamental group for $n<6$ and describe its homology for $n=6,7$. For arbitrary $n$, we compute its first homology and its Euler characteristic. We use three geometric approaches. On one hand, $Q_n$ is naturally defined as the complement to an arrangement of codimension-2 subtori in a real torus. On the other hand, $Q_n$ is homotopy equivalent to an explicit nonpositively curved cubical complex. Finally, $Q_n$ can also be assembled from no-3-equal manifolds of the real line. We also observe that, up to homotopy, $Q_n$ may be identified with a subspace of the oriented double cover of the moduli space $\overline{\mathcal{M}}_{0,n}(\mathbb{R})$ of stable real rational curves with $n$ marked points. This gives an embedding of $π_1 Q_n$ into the pure cactus group. As a corollary, we see that $π_1 Q_n$ is residually nilpotent.

Figures (4)

• Figure 1: The graph of groups $\Phi_n$ for $n=7$.
• Figure 2: A point in $\overline{\mathcal{M}}_{0,10}(\mathbb R)$ coming from ${Q}_{10}$.
• Figure 3: Points in $\overline{\mathcal{M}}_{0,6}(\mathbb R)$ not coming from $M_{6}{(S^1)}$.
• Figure 4: The generators $\zeta$ and $\sigma_2$ of $\Upsilon_5$.

Theorems & Definitions (24)

• Definition 1
• Theorem 2
• proof
• Example 3
• Theorem 4
• proof
• Theorem 5
• Theorem 6
• Theorem 7
• proof : Proof of Theorem \ref{['les']}