Quotients of torus endomorphisms have parabolic orbifolds.

An orientation-preserving branched covering map f : S 2 → S 2 is a Thurston map if deg(f) ≥ 2 and the orbit of each critical point is finite. Associated to each Thurston map f, there is an orbifold Of . This work investigates the relationship between f and Of . A map f is said to be a quotient of a torus endomorphism (QOTE) if there exists a map F : T 2 → T 2 with deg(F) ≥ 2 and a branched covering map π : T 2 → S 2 such that f ◦ π = π ◦ F. Note that every QOTE map f is a Thurston map. This work answers a question posed by M. Bonk and D. Meyer [Arnold Math. J. 6, No. 3–4, 495–521 (2020; Zbl 1486.30077)]: Does the orbifold Of have vanishing Euler characteristic when f : S 2 → S 2 is QOTE? The answer is yes. Building 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 7→ Ax+b, det(A) > 1. Following [loc. cit.], the authors prove that the following statements are equivalent: • f is a Thurston map with χ(Of ) = 0 and no periodic critical points; • f is Thurston equivalent to a Lattès-type map; • f is Thurston equivalent to a quotient of a torus endomorphism