Three-dimensional maps and subgroups growth

Pavages tridimensionnels et croissance de sous-groupes

Laura Ciobanu (1) , Alexander Kolpakov (2)

1 MATHEMATICS DEPARTMENT OF HERIOT-WATT UNIVERSITY - School of Mathematical and Computer Sciences
2 Laboratoire de Mathématiques de Neuchâtel

In this paper we derive a generating series for the number of free subgroups of finite index in ∆ + = Z 2 * Z 2 * Z 2 by using the bijection between free subgroups of index n in ∆ + and cellular complexes known as pavings or three-dimensional maps, on n darts, and show that this generating series is non-holonomic. We then provide and study the generating series for conjugacy classes of free subgroups of finite index in ∆ + , which correspond to isomorphism classes of pavings. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, and the equivalent types of pavings and their isomorphism classes.

Mots clés: subgroup growth; growth series; free product; free group; map; paving

Mots clés: série de Poincaré; dessin d'enfant; produit libre; groupe libre; croissance de sous-groupes; pavage

Domaines: Combinatoire [math.CO]; Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Mathématique discrète [cs.DM]; Théorie des groupes [math.GR]; Théorie et langage formel [cs.FL]; Topologie algébrique [math.AT]; Topologie géométrique [math.GT]

Laura Ciobanu, Alexander Kolpakov. Three-dimensional maps and subgroups growth. 2019. ⟨hal-01858014v2⟩