A bijection for rooted maps on general surfaces (extended abstract)

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent ^5(h₂⁻¹⁾ for both orientable and non-orientable maps of Euler characteristic 2 − 2h and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor n¹/⁴ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size n on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.