Séminaire Lotharingien de Combinatoire, 93B.143 (2025), 12 pp.
Jake Levinson and Haggai Liu
Fundamental Groups of Moduli Spaces of Real Weighted Stable Curves
Abstract.
The ordinary and Sn-equivariant fundamental groups of the moduli space M-0,n+1(R) of real (n+1)-marked stable curves of genus 0 are known as cactus groups Jn and have applications both in geometry and the representation theory of Lie algebras.
In this paper, we compute the ordinary and Sn-equivariant fundamental groups of the Hassett space of weighted real stable curves M-0,A(R) with Sn-symmetric weight vector A = (1/a, ..., 1/a, 1), which we call weighted cactus groups Jna. We show that Jna is obtained from the usual cactus presentation by introducing braid relations, which successively simplify the group from Jn to Sn ⋊ Z2 as a increases.
Our proof is by decomposing M-0,A(R) as a polytopal complex, generalizing a similar known decomposition for M-0,n+1(R). In the unweighted case, these cells are known to be cubes and are `dual' to the usual decomposition into associahedra. For M-0,A(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.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
The following versions are available: