Algebraic Graph Theory Seminar - Martin Škoviera (12.10.2018)
Friday 12.10.2018 at 13:30, Lecture room M/XI
By: Martin Mačaj
Complete regular dessins and skew-morphisms of cyclic groups
A dessin d'enfant (dessin, for short) is a 2-cell embedding of a connected bipartite graph into a closed oriented surface, endowed with a fixed vertex 2-colouring. A dessin is regular if its automorphism group acts regularly on the edges. Dessins were introduced by Grothendieck over thirty years ago as a combinatorial counterpart of algebraic curves. In this talk we deal with regular dessins whose underlying graph is a complete bipartite graph Km,n, called (m,n)-complete dessins. We discuss a rather surprising correspondence between (m,n)-complete regular dessins and skew-morphisms of cyclic groups. We show that every (m,n)-complete regular dessin D gives rise to a closely related ``reciprocal'' pair of skew-morphisms of Z_n and Z_m. Conversely, D can be easily reconstructed from the reciprocal pair up to isomorphism. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and reciprocal pairs of skew-morphisms of cyclic groups are all in a one-to-one correspondence. We apply the main result to determining all pairs of integers m and n for which there exists, up to interchange of colours, exactly one (m,n)-complete regular dessin. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order m and n is abelian, which eventually comes down to the condition (m,\phi(n))=(\phi(m),n)=1.
(This is a joint work with Y.-Q. Feng, K. Hu, N.-E. Wang, and R. Nedela.)