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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 $ arphi$ of a finite group $A$ is a permutation on $A$ such that $ arphi(1)=1$ and $ arphi(xy)= arphi(x) arphi^{pi(x)}(y)$ for all $x,yin A$ where $pi:A omathbb{Z}_{| arphi|}$ 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 Skoviera (2008). This is joint work with Y.Q.Feng, Kan Hu and M. { S}koviera.
Skew morphisms of cyclic groups and complete regular dessins - TIB AV-Portal