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 Δ ⁺= Z₂∗ Z₂∗ Z₂ via a simple bijection between pavings and finite index subgroups which can be deduced from the action of Δ ⁺ 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 Δ ⁺. Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on n≤ 16 darts.