The moduli space of cactus flower curves and the virtual cactus group

The space \( \ft_n = \C^n/\C \) of \(n\) points on the line modulo translation has a natural compactification \( \overline \ft_n \) as a matroid Schubert variety. In this space, pairwise distances between points can be infinite; it is natural to imagine points at infinite distance from each other as living on different projective lines. We call such a configuration of points a ``flower curve'', since we picture the projective lines joined into a flower. Within \( \ft_n \), we have the space \( F_n = \C^n \setminus \Delta / \C \) of \( n\) distinct points. We introduce a natural compatification \( \overline F_n \) along with a map \( \overline F_n \rightarrow \overline \ft_n \), whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both \(\overline \ft_n\) and \(\overline F_n\), are special fibers of \(1\)-parameter families whose generic fibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces of stable genus \(0\) curves with \(n+2\) marked points. We find combinatorial models for the real loci \( \overline \ft_n(\BR) \) and \( \overline F_n(\BR) \). Using these models, we prove that these spaces are aspherical and that their equivariant fundamental groups are the virtual symmetric group and the virtual cactus groups, respectively. The degeneration of a twisted real form of the Deligne-Mumford space to \(\overline F_n(\mathbb{R})\) gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group.