Trivalent Diagrams, Modular Group and Triangular Maps

The aim of the paper is to give a preliminary overview of some of the results of the thesis prepared by the author. We propose a bijective classification of the subgroups of the modular group by pointed trivalent diagrams. Conjugacy classes of those subgroups are in one to one correspondence with unpointed trivalent diagrams. We also give in the form of generating series, the number of those trivalent diagrams both pointed or not, as well as generating series for both rooted and unrooted versions of triangular maps up to isomorphism. That later results was a difficult open problem. Introduction The theory of combinatorial maps is a venerable subject dating back to Cayley and Hamilton. Since those time it generated an impressive amount of results of all sorts concerning the enumeration problem of counting the rooted combinatorial maps. Those results came from various communities of researchers, each with its own methods and tradition. Among them, enumerative combinatorists of course played a significant role, starting with pioneering works by Tutte [16] on rooted planar maps. Theoretical physicists also played a significant role, starting with the work by t’Hooft [15] on integrals on random matrix spaces. Pure mathematicians like Harer and Zagier [7] also have contributed to the theory in connection with cutting edge algebraic geometry problems concerning modular spaces of Riemann surfaces. Last but not least, one must mention in mathematical physics the Witten-Kontsevich model of quantum gravity [9] using in a central fashion the higher combinatorics of triangular maps and trivalent diagrams. Although a lot is known concerning the theory of rooted combinatorial maps, very little is known concerning the outstanding problem of enumeration of unrooted combinatorial maps up to isomorphism. It appears as a very difficult problem of combinatorics, which stayed barely untouched for almost 150 years. As a matter of fact, the only general result on those important objects were up to now contained in the recent paper by A. Mednykh