Joining Dessins Together

Realisation of groups as automorphism groups in permutational categories

“…This is illustrated in Figure 5, where the (1)-handle at the bottom of G is joined to that at the top of H by two dashed edges to form G(1)H; these edges can be carried by a tube connecting the two surfaces, showing the additivity of the genera (both equal to 0 here). (Further details about this and more general joining operations on dessins can be found in [26].) Figure 5, to that at the bottom of the next map G. In this chain, each copy of G has an unused (1)-handle; we join these arbitrarily in pairs, using (1)-compositions.…”