Three-dimensional maps and subgroup growth

TL;DR

This paper derives a generating series for three-dimensional maps called pavings, counts free subgroups of a specific group via a bijection, and explores their properties and statistics through computational experiments.

Contribution

It introduces a generating series for pavings in three dimensions, linking them to subgroup counts and analyzing their properties, including non-holonomicity.

Findings

Derived a generating series for pavings on n darts.

Established a bijection between pavings and subgroups of a free product of cyclic groups.

Performed computational experiments on pavings with up to 16 darts.

Abstract

In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on $n$ darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index $n$ in $\Delta^+ = \mathbb{Z}_2*\mathbb{Z}_2*\mathbb{Z}_2$ via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $\Delta^+$ on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in $\Delta^+$. Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $n\leq 16$ darts.