Planar Bilipschitz Extension from Separated Nets

We prove that every $L$-bilipschitz mapping ${\mathbb{Z}}^2\to {\mathbb{R}}^2$ can be extended to a $C(L)$-bilipschitz mapping ${\mathbb{R}}^2\to {\mathbb{R}}^2$ and provide a polynomial upper bound for $C(L)$. Moreover, we extend the result to every separated net in ${\mathbb{R}}^2$ instead of ${\mathbb{Z}}^2$, with the upper bound gaining a polynomial dependence on the separation and net constants associated to the given separated net. This answers an Oberwolfach question of Navas from 2015 and is also a positive solution of the two-dimensional form of a decades old open (in all dimensions at least two) problem due to Alestalo, Trotsenko and Väisälä.