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 V\"ais\"al\"a.