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Complete regular dessins and skew-morphisms of cyclic groups

Banff International Research Station for Mathematical Innovation and Discovery

In this talk we introduce a surprising correspondence between $(m,n)$-complete regular dessins and admissible pairs of skew-morphisms of the cyclic groups of orders $m$ and $n$. A skew-morphism $\varphi$ of a finite group $A$ is a permutation on $A$ such that $\varphi(1)=1$ and $\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$ for all $x,y\in A$ where $\pi:A\to\mathbb{Z}_{|\varphi|}$ is an integer function. We determine the pairs $(m,n)$ for which there exists exactly one dual pair of $(m,n)$-complete regular dessins, thus generalising an earlier result by Jones, Nedela and \v Skoviera (2008). This is joint work with Y.Q.Feng, Kan Hu and M. {\v S}koviera.

Mathematics; Manifolds and cell complexes; Group theory and generalizations; Discrete mathematics

video/mp4

Author affiliation: University of West Bohemia

2018-04-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65380

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/