Generalizations of the groups $G_{n}^{k}$: graphs, moduli spaces, algebraic geometry, spherical braids

Abstract

In this work, we construct a generalization of the $G_n^k$-theory to the case of an arbitrary hypergraph. The case of spherical braids is considered separately, using the stratification of the moduli space $\mathcal{M}_n(S^2)$ and the hypergraph $\Gamma_n^{\mathrm{sph}}$ encoding projective constraints. In contrast to the original $G_{n}^{k}$ theory, where codimension-one properties are determined by exactly $k$ particles, the present work considers various cases corresponding to strata of codimension~$1$. These groups admit nice maps to free products of cyclic groups. Among unsolved problems, we emphasize the question how the above construction works for abelian varieties and, in particular, for elliptic curves.