On an integral representation for the genus series for 2-cell embeddings

Jackson, D. M.

doi:10.1090/S0002-9947-1994-1236224-5

On an integral representation for the genus series for $2$-cell embeddings
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Trans. Amer. Math. Soc. 344 (1994), 755-772 Request permission

Abstract:

An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, $\mathbb {C}{\mathfrak {G}_n}$, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals.

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