Abstract We study problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in $${\mathbb {T}}:=\{z\in {\mathbb {C}}:|z|=1\}$$ T : = { z ? C | ...

We give a characterization of metric spaces quasisymmetrically equivalent to finitely connected circle domain. This result generalizes the uniformization Ahlfors 2-regular by Merenkov and Wildrick.

We show that for each cusp on the boundary of Maskit’s embedding M ⊂ H of the Teichmüller space of punctured tori there is a sequence of parameters in the complement ofM converging to the cusp such that the parameters correspond to discrete groups with elliptic elements. Using Tukia’s version of Marden’s isomorphism theorem we identify them as cusps on the boundary of certain deformation spaces...