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Fundamental groups of moduli spaces of real weighted stable curves

Jake Levinson, Haggai Liu

TL;DR

This work computes the ordinary and $S_n$-equivariant fundamental groups of the real Hassett moduli spaces $\overline{M_{0,\mathcal{A}(a)}}(\mathbb{R})$, introducing the weighted cactus groups $J_n^a$ that interpolate between the ordinary cactus group and a semidirect product $S_n\rtimes\mathbb{Z}/2\mathbb{Z}$ as $a$ increases. The authors construct a dual polytopal decomposition of these spaces, revealing that cells are products of permutahedra indexed by $a$-stable trees, and use the $2$-skeleton to derive explicit presentations for $J_n^a$ with generators $s_{p,q}$ constrained by $q-p\in\{1\}\cup\{a,...,n-1\}$ and relations including involution, commuting, cactus, and generalized braid relations. The approach unifies real moduli geometry with combinatorial invariants via a weighted distance algorithm and a dual CW-structure, yielding a clear pathway from the unweighted cactus group $J_n$ to its weighted variants. This provides a detailed understanding of the algebraic structure of real weighted moduli spaces, with implications for representation theory, geometry, and combinatorics.

Abstract

The ordinary and $S_n$-equivariant fundamental groups of the moduli space $\overline{M_{0,n+1}}(\mathbb{R})$ of real $(n+1)$-marked stable curves of genus $0$ are known as \emph{cactus groups} $J_n$ and have applications both in geometry and the representation theory of Lie algebras. In this paper, we compute the ordinary and $S_n$-equivariant fundamental groups of the Hassett space of weighted real stable curves $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$ with $S_n$-symmetric weight vector $\mathcal{A} = (1/a, \ldots, 1/a, 1)$, which we call \emph{weighted cactus groups} $J_n^a$. We show that $J_n^a$ is obtained from the usual cactus presentation by introducing braid relations, which successively simplify the group from $J_n$ to $S_n \rtimes \mathbb{Z}/2\mathbb{Z}$ as $a$ increases. Our proof is by decomposing $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$ as a polytopal complex, generalizing a similar known decomposition for $\overline{M_{0,n+1}}(\mathbb{R})$. In the unweighted case, these cells are known to be cubes and are `dual' to the usual decomposition into associahedra (by the combinatorial type of the stable curve). For $\overline{M_{0,\mathcal{A}}}(\mathbb{R})$, our decomposition instead consists of products of permutahedra. The cells of the decomposition are indexed by weighted stable trees, but `dually' to the usual indexing.

Fundamental groups of moduli spaces of real weighted stable curves

TL;DR

This work computes the ordinary and -equivariant fundamental groups of the real Hassett moduli spaces , introducing the weighted cactus groups that interpolate between the ordinary cactus group and a semidirect product as increases. The authors construct a dual polytopal decomposition of these spaces, revealing that cells are products of permutahedra indexed by -stable trees, and use the -skeleton to derive explicit presentations for with generators constrained by and relations including involution, commuting, cactus, and generalized braid relations. The approach unifies real moduli geometry with combinatorial invariants via a weighted distance algorithm and a dual CW-structure, yielding a clear pathway from the unweighted cactus group to its weighted variants. This provides a detailed understanding of the algebraic structure of real weighted moduli spaces, with implications for representation theory, geometry, and combinatorics.

Abstract

The ordinary and -equivariant fundamental groups of the moduli space of real -marked stable curves of genus are known as \emph{cactus groups} and have applications both in geometry and the representation theory of Lie algebras. In this paper, we compute the ordinary and -equivariant fundamental groups of the Hassett space of weighted real stable curves with -symmetric weight vector , which we call \emph{weighted cactus groups} . We show that is obtained from the usual cactus presentation by introducing braid relations, which successively simplify the group from to as increases. Our proof is by decomposing as a polytopal complex, generalizing a similar known decomposition for . In the unweighted case, these cells are known to be cubes and are `dual' to the usual decomposition into associahedra (by the combinatorial type of the stable curve). For , our decomposition instead consists of products of permutahedra. The cells of the decomposition are indexed by weighted stable trees, but `dually' to the usual indexing.

Paper Structure

This paper contains 22 sections, 18 theorems, 109 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Acknowledgments
  4. Setup and Background
  5. Stable and $a$-stable trees
  6. Weighted stable trees
  7. Curves and moduli spaces
  8. Symmetric weights
  9. $G$-equivariant Fundamental Group
  10. Cactus groups as $S_n$-equivariant fundamental groups
  11. Dual CW-Structure on the Hassett Space and its Double Cover
  12. Dual Stratification of $\overline{M_{0,n+1}}(\mathbb{R})$
  13. The distance algorithm
  14. From marked curves to trees
  15. Dual cubes and refined cubes
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $a\ge 3$. The weighted cactus group $J_n^a$ has generators $s_{p,q}, 1\le p<q\le n$ satisfying either $q-p\in \{a,a+1, \ldots, n-1\}$ or $q-p=1$, with the relations:

Figures (12)

  • Figure 1: A local picture of the blowdown map $\overline{M_{0,{\mathcal{A}(2)}}}(\mathbb{R}) \to \overline{M_{0,{\mathcal{A}(3)}}}(\mathbb{R})$, with $5$ marked points. Left: three square regions homeomorphic to $[-1, 1] \times \Pi_2$, each indexed by a $2$-stable tree. Note that on the inner hexagon, antipodal points are identified, giving a copy of $\mathbb{RP}^1$ (the exceptional divisor). Right: Contracting the inner boundary to a point and merging the three cells yields a copy of $\Pi_3 \subseteq \overline{M_{0,{\mathcal{A}({3})}}}(\mathbb{R})$ indexed by a $3$-stable tree. On the right, we also show vertex labels and one illustrative edge label, which do not change. The interior edges on the left are labeled by $2$-stable trees whose $a$-compression is the $3$-stable tree labeling the entire $\Pi_3$: , , .
  • Figure 2: Example of tree compression. Observe that $\varpi_7 = \varpi_{7,3}\circ \varpi_3$.
  • Figure 3: ($n=10$) Left: A stable tree $\tau\in \mathop{\mathrm{StRtree}}\nolimits([10])$ with the root colored red and internal edges coloured blue. Right: An element of the standard cell $X_{\tau}\subseteq \overline{M_{0,11}}(\mathbb{R})$ with $\tau$ overlayed. Here, $\dim X_{\tau}=(4-3)+(5-3)+(4-3)+(4-3)=5$.
  • Figure 4: The path $\hat{s}_{p,q}$ in $\overline{M_{0,{n+1}}}(\mathbb{R})$ corresponding to $s_{p,q}\in J_n$: the $p$-th through $q$-th marked points approach one another and collide, reversing their order. (The positions of the marked points are not drawn to scale.)
  • Figure 5: Left: The tree $\tau^{\text{rfd}}(0, 1, 2, 5, 6, 6.5)$, obtained by running the distance algorithm on the successive differences (Def. \ref{['def:dist-alg-diff-to-rtree']}). Right: A stable curve $(C; x_{\bullet})$ shown approximately to scale, with the tree $\tau^{\text{rfd}}(C; x_{\bullet})$ overlaid. Each component is coordinatized by setting the point nearest to $x_{n+1}$ to $\infty$ and the tree on each component is calculated as in the distance algorithm. These trees are attached along the three highlighted edges shown in blue. The resulting tree is only well-defined up to flipping each irreducible component, i.e. flipping at the child of each highlighted edge.
  • ...and 7 more figures

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1: Presentations of $PJ_n^a$
  • Theorem 1.2
  • Remark 2: Double Covers
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • ...and 42 more