  We study large uniform random quadrangulations whose genus grow linearly with
the number of faces, whose local convergence was recently established by
Budzinski and the author arXiv:1902.00492,arXiv:2012.05813. Here we study
several properties of these objects which are not captured by the local
topology. Namely we show that balls around the root are planar whp up to
logarithmic radius, and we prove that there exists short non-contractible
cycles with positive probability.
