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Abstract

The purpose of this paper is to prove that proper holomorphic self-mappings of the minimal ball are biholomorphic. The proof uses the scaling technique applied at a singular point and relies on the fact that a proper holomorphic mapping $f: D\rightarrow {\mit \Omega }$ with branch locus $V_f$ is factored by automorphisms if and only if $f_{*}(\pi _1(D\setminus f^{-1} (f(V_f)), x))$ is a normal subgroup of $\pi _1({\mit \Omega } \setminus f(V_f), b)$ for some $b\in {\mit \Omega } \setminus f(V_f)$ and $x\in f^{-1}(b)$.