Relative moduli spaces of complex structures: an example

Let M and N be even-dimensional oriented real manifolds, and $u:M \to N$ be a smooth mapping. A pair of complex structures at M and N is called u-compatible if the mapping u is holomorphic with respect to these structures. The quotient of the space of u-compatible pairs of complex structures by the group of u-equivariant pairs of diffeomorphisms of M and N is called a moduli space of u-equivariant complex structures. The paper contains a description of the fundamental group G of this moduli space in the following case: $N = CP^1, M \subset CP^2$ is a hyperelliptic genus g curve given by the equation $y^2 = Q(x)$ where Q is a generic polynomial of degree 2g+1, and $u(x,y) = y^2$. The group G is a kernel of several (equivalent) actions of the braid-cyclic group $BC_{2g}$ on 2g strands. These are: an action on the set of trees with 2g numbered edges, an action on the set of all splittings of a (4g+2)-gon into numbered nonintersecting quadrangles, and an action on a certain set of subgroups of the free group with 2g generators. $G_{2g} \subset BC_{2g}$ is a subgroup of the index $(2g+1)^{2g-2}$. Key words: Teichm\"uller spaces, Lyashko-Looijenga map, braid group.