Quotients of torus endomorphisms have parabolic orbifolds (Q6580055)

An orientation-preserving branched covering map \(f: \mathbb{S}^2 \to \mathbb{S}^2\) is a Thurston map if \(\deg(f) \geq 2\) and the orbit of each critical point is finite. Associated to each Thurston map \(f\), there is an orbifold \(\mathcal{O}_f\). This work investigates the relationship between \(f\) and \(\mathcal{O}_f\).\N\NA map \(f\) is said to be a quotient of a torus endomorphism (QOTE) if there exists a map \(F: \mathbb{T}^2 \to \mathbb{T}^2\) with \(\deg(F) \geq 2\) and a branched covering map \(\pi: \mathbb{T}^2 \to \mathbb{S}^2\) such that \(f \circ \pi = \pi \circ F\). Note that every QOTE map \(f\) is a Thurston map.\N\NThis work answers a question posed by \textit{M. Bonk} and \textit{D. Meyer} [Arnold Math. J. 6, No. 3--4, 495--521 (2020; Zbl 1486.30077)]: Does the orbifold \(\mathcal{O}_f\) have vanishing Euler characteristic when \(f: \mathbb{S}^2 \to \mathbb{S}^2\) is QOTE? The answer is yes.\N\NBuilding on this result, the authors examine the connections between Thurston maps and Lattès-type maps. A QOTE is Lattès-type if we can take \(F\) affine of the form \(x\mapsto Ax+b\), \(\det(A)>1\). Following [loc. cit.], the authors prove that the following statements are equivalent:\N\begin{itemize}\N\item \(f\) is a Thurston map with \(\chi(\mathcal O_f)=0\) and no periodic critical points;\N\item \(f\) is Thurston equivalent to a Lattès-type map;\N\item \(f\) is Thurston equivalent to a quotient of a torus endomorphism.\N\end{itemize}