On hyperbolic cobordisms and Hurwitz classes of holomorphic coverings

Abstract

In this article we show that for every collection C \mathcal {C} of an even number of polynomials, all of the same degree d > 2 d>2 and in general position, there exist two hyperbolic 3 3 -orbifolds M 1 M_1 and M 2 M_2 with a Möbius morphism α : M 1 → M 2 \alpha :M_1\rightarrow M_2 such that the restriction of α \alpha to the boundaries ∂ M 1 \partial M_1 and ∂ M 2 \partial M_2 forms a collection of maps Q Q in the same conformal Hurwitz class of the initial collection C \mathcal {C} . Also, we discuss the relationship between conformal Hurwitz classes of rational maps and classes of continuous isomorphisms of sandwich products on the set of rational maps.