Alberti's rank one theorem and quasiconformal mappings in metric measure spaces

We investigate a version of Alberti’s rank one theorem in Ahlfors regular metric spaces, as well as a connection with quasiconformal mappings. More precisely, we give a proof of the rank one theorem that partially follows along the usual steps, but the most crucial step consists in showing for $f\in BV(X;Y)$ that at $\mathrm{\parallel }Df{\mathrm{\parallel }}^s$-a.e. $x\in X$, the mapping $f$ ``behaves non-quasiconformally’'.