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Three-dimensional maps and subgroup growth.

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Summary
This summary is machine-generated.

This study introduces a generating series for 3D maps (pavings) and free subgroups, solving a 3D analogue of Tutte's problem. The series reveals insights into their combinatorial and topological properties.

Keywords:
05E4514N1020E0720H1033C20

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Area of Science:

  • Combinatorics
  • Algebraic Topology
  • Group Theory

Background:

  • Cellular complexes, known as pavings or 3D maps, are studied in combinatorics.
  • Tutte's problem, concerning planar maps and their generating series, is a significant area of research.
  • Free subgroups of finite index in the Grigorchuk group () have connections to various mathematical fields.

Purpose of the Study:

  • To derive a generating series for 3D maps (pavings) on n darts.
  • To establish a bijection between pavings and free subgroups of index n in the Grigorchuk group.
  • To analyze the properties of these generating series, including their non-holonomic nature, and their connection to conjugacy classes.

Main Methods:

  • Derivation of generating series using combinatorial techniques.
  • Establishing a bijection between pavings and free subgroups via group action on cosets.
  • Computational experiments using custom software to analyze paving statistics.

Main Results:

  • A generating series for pavings (3D maps) on n darts was derived.
  • This series also counts free subgroups of index n in the Grigorchuk group.
  • The generating series was shown to be non-holonomic.
  • A generating series for isomorphism classes of pavings was provided, corresponding to conjugacy classes of free subgroups.

Conclusions:

  • The study successfully extends Tutte's problem to three dimensions using generating series for pavings.
  • A direct link between 3D maps and free subgroups of the Grigorchuk group was demonstrated.
  • Computational results offer preliminary insights into the combinatorial and topological characteristics of these structures.