Three-dimensional maps and subgroup growth

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 ${\mathrm{\Delta }}^{+}={\mathbb{Z}}_2\ast {\mathbb{Z}}_2\ast {\mathbb{Z}}_2$ via a simple bijection between pavings and finite index subgroups which can be deduced from the action of ${\mathrm{\Delta }}^{+}$ 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 ${\mathrm{\Delta }}^{+}$. Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $n\le 16$ darts.