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

TL;DR

This paper investigates the growth rates of canonical groups associated with face 2-coloured triangulations of the sphere, establishing bounds on the maximal order of their torsion subgroups through connections with directed Eulerian digraphs.

Contribution

It introduces a novel relationship between face 2-coloured spherical triangulations and directed Eulerian digraphs to derive bounds on group growth rates.

Findings

Established improved bounds for the growth rate of the maximal order of the canonical group.

Connected triangulation groups with Eulerian digraph sand-pile groups to analyze asymptotic behavior.

Provided mathematical framework for understanding group growth in face 2-coloured triangulations.

Abstract

Let $\mathcal{G}$ be a properly face 2-coloured (say black and white) \break piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$ where $\mathcal{C}$ is a finite abelian group. The finite torsion subgroup $\mathcal{C}$ is referred to as the canonical group of the triangulation. Let $m_t$ be the maximal order of $\mathcal{C}$ over all properly face two-coloured spherical triangulations with $t$ triangles of each colour. By relating properly face two-coloured spherical triangulations to directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to $\mathcal{C}$ we provide improved upper and lower bounds for $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.