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.