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 nle 16 darts.
Highlights
In this note we explore the correspondence between the number of rooted threedimensional maps, or pavings, on n darts, as introduced in [2,19,38], and free subgroups of given index n in the free product + = Z2 ∗ Z2 ∗ Z2, in order to obtain generating series, new formulas and asymptotics for these objects
While similar connections between free subgroups of finite index in certain Fuchsian triangle groups and two-dimensional maps have been previously exploited by a number of authors [7,16,22,23,24,25,32,39,40,42], relatively little has been done for maps in 3 dimensions; this paper is a step towards developing the theory and computation in higher dimensions
Throughout the paper we give several concrete and illustrative examples, and in Sect. 7 we provide some statistical information about pavings on n ≤ 16 darts using a computer program Nem [6] created for the purpose of their enumeration and classification
Summary
There is a natural way to associate with every free subgroup of index n in + a paving on n darts, and we give new quantitative information, as well as examples with concrete computations, for both kinds of objects, the geometric ones and the. While similar connections between free subgroups (and their conjugacy classes) of finite index in certain Fuchsian triangle groups and two-dimensional maps have been previously exploited by a number of authors [7,16,22,23,24,25,32,39,40,42], relatively little has been done for maps in 3 dimensions; this paper is a step towards developing the theory and computation in higher dimensions. Throughout the paper we give several concrete and illustrative examples, and in Sect. 7 we provide some statistical information about pavings on n ≤ 16 darts using a computer program Nem [6] created for the purpose of their enumeration and classification
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