On hyperbolic cobordisms and Hurwitz classes of holomorphic coverings

TL;DR

This paper explores the existence of hyperbolic 3-orbifolds related to collections of polynomials and their boundary maps within the same conformal Hurwitz class, linking complex dynamics and geometric structures.

Contribution

It establishes the existence of hyperbolic orbifolds with boundary maps in the same Hurwitz class as given polynomial collections, connecting complex dynamics with hyperbolic geometry.

Findings

Existence of hyperbolic 3-orbifolds with boundary maps in the same Hurwitz class as polynomial collections.

Relationship between conformal Hurwitz classes of rational maps and isomorphisms of sandwich products.

Insights into the interplay between polynomial collections, orbifolds, and conformal classes.

Abstract

In this article we show that for every collection $\mathcal{C}$ of an even number of polynomials, all of the same degree $d>2$ and in general position, there exist two hyperbolic $3$-orbifolds $M_1$ and $M_2$ with a M\"obius morphism $\alpha:M_1\rightarrow M_2$ such that the restriction of $\alpha$ to the boundaries $\partial M_1$ and $\partial M_2$ forms a collection of maps $Q$ in the same conformal Hurwitz class of the initial collection $\mathcal{C}$. 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.