Growth rates of groups associated with face 2-coloured triangulations\n and directed Eulerian digraphs on the sphere

2014/08/13 by Thomas A. McCourt, McCourt, Thomas A.
Computer Science · Mathematics · #05B15 #05C10 #05C20 #05C25 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1408.2984

openalex publication_date 2014/08/13 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Let \G be a properly face 2-coloured (say black and white) break\npiecewise-linear triangulation of the sphere with vertex set V. Consider the\nabelian group \A_W generated by the set V, with relations\nr+c+s=0 for all white triangles with vertices r, c and s. The group\n\A_B can be defined similarly, using black triangles. These groups\nare related in the following manner\n\A_W\≅\A_B\≅\ℤ\⊕\ℤ\⊕\C\nwhere \C is a finite abelian group.\n The finite torsion subgroup \C is referred to as the canonical\ngroup of the triangulation. Let m_t be the maximal order of \C\nover all properly face two-coloured spherical triangulations with t triangles\nof each colour. By relating properly face two-coloured spherical triangulations\nto directed Eulerian spherical embeddings of digraphs whose abelian sand-pile\ngroups are isomorphic to \C we provide improved upper and lower\nbounds for \lim \sup_t\→\∞(m_t)^1/t.\n

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