Triangulations of the sphere, bitrades and abelian groups

2011/12/22 by Simon R. Blackburn⋆, Blackburn, Simon R., Thomas A. McCourt +1
Computer Science · Mathematics · #05B07 #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1112.5423

openalex publication_date 2011/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a triangulation of the sphere with vertex set V, such that the faces of the triangulation are properly coloured black and white. Motivated by applications in the theory of bitrades, Cavenagh and Wanless defined A_W to be the abelian group generated by the set V, with relations r+c+s=0 for all white triangles with vertices r, c and s. The group A_B can be defined similarly, using black triangles. The paper shows that A_W and A_B are isomorphic, thus establishing the truth of a well-known conjecture of Cavenagh and Wanless. Connections are made between the structure of A_W and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given. The relevance of the group A_W to the understanding of the embeddings of a partial latin square in an abelian group is also explained.

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