Quasisymmetric uniformization of cellular branched covers

Abstract: In this paper, we introduce a class of branched covering maps on topological $n$-spheres $\mathcal{S}^n$ called expanding cellular Markov branched covers which might be viewed as analogs of expanding Thurston maps. Similarly as expanding Thurston maps, expanding cellular Markov branched covers admit visual metrics. In this paper, we study the quasisymmetric uniformization problem corresponding to visual metrics of expanding cellular Markov branched covers. More precisely, for a cellular Markov branched cover $f\colon \mathbb{S}^n\to \mathbb{S}^n$ and a visual metric $\varrho$ of $f$, we prove that $\varrho$ is quasisymmetrically equivalent to the chordal metric if and only if $f$ is uniformly quasiregular, and the local multiplicity of the family ${f^m}_{m\in\mathbb{N}}$ is bounded uniformly.