Three-dimensional maps and subgroup growth

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 ${\Delta }^{+}={\mathbb{Z}}_2\ast {\mathbb{Z}}_2\ast {\mathbb{Z}}_2$ 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 $n\le 16$ darts.

17 pages, 6 figures, 1 table; auxiliary files on GitHub: https://github.com/bottine/nem and https://github.com/sashakolpakov/monty-3d

References in corpus (2)

• Sur la Classification et le Denombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison
• Free subgroups of free products and combinatorial hypermaps

Cited by in corpus (1)

• Statistics of subgroups of the modular group