Statistics of isomorphism types in free products

Müller, Thomas; Schlage-Puchta, Jan Christoph

doi:10.1016/j.aim.2009.12.011

Cited by 9 publications

(6 citation statements)

References 11 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After precomputing the coefficients of all EGSs under consideration, we can use Equations (22) and (31) to build a random sampler. By Equation ( 22), the graph is not loop-free with probability [z n ]G…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…In 2004, Müller and Schlage-Puchta [29] studied the isomorphism types of finite index subgroups of free products of cyclic groups (see also [31], and [30] for an extension to Fuchsian groups). Let G = C * e 1 p 1 * • • • * C * et pt * F r , where C q is the cyclic group of order q, the p i are pairwise distinct, the superscript * e means the free product of e copies, and F r is the rank r free group.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Statistics of subgroups of the modular group

Bassino,

Nicaud,

Weil

2020

Preprint

View full text Add to dashboard Cite

We count the finitely generated subgroups of the modular group PSL 2 (Z). More precisely: each such subgroup H can be represented by its Stallings graph Γ(H), we consider the number of vertices of Γ(H) to be the size of H and we count the subgroups of size n. Since an index n subgroup has size n, our results generalize the known results on the enumeration of the finite index subgroups of PSL 2 (Z). We give asymptotic equivalents for the number of finitely generated subgroups of PSL 2 (Z), as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size n subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size n subgroup (resp. finite index subgroup, free subgroup) of PSL 2 (Z). Contents

show abstract

Section: ⊓ ⊔mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Statistics of subgroups of the modular group

Bassino,

Nicaud,

Weil

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…General subgroup growth is the subject of the book [20], and further information on subgroup growth in free products of cyclic groups can be found in [3,[26][27][28][29][30]40]. There, the general theory of subgroup structure in free products of (finite and infinite) cyclic groups is enhanced by using the methods of representation theory, analytic number theory and probability theory, among other tools.…”

Section: Introductionmentioning

confidence: 99%

Three-dimensional maps and subgroup growth

2021

View full text Add to dashboard Cite

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 $$\Delta ^+ = {\mathbb {Z}}_2*{\mathbb {Z}}_2*{\mathbb {Z}}_2$$ Δ + = Z 2 ∗ Z 2 ∗ Z 2 via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $$\Delta ^+$$ 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 $$\Delta ^+$$ Δ + . Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $$n\le 16$$ n ≤ 16 darts.

show abstract

“…Date: February 13, 2019. 1 also known as ribbon graphs or "fat" graphs 1 General subgroup growth is the subject of the monograph [21] by Lubotzky and Segal, and further information on counting the number of subgroups in free products of cyclic groups of prime orders can be found in the papers by Müller and Schlage-Puchta [27,28,29]. There they employ the general theory of subgroup structure in free products of (finite and infinite) cyclic groups together with representation theory, analytic number theory and probability theory, among other tools.…”

Section: Introductionmentioning

confidence: 99%

Free subgroups of free products and combinatorial hypermaps

Ciobanu

Kolpakov

2019

Discrete Mathematics

View full text Add to dashboard Cite

We derive a generating series for the number of free subgroups of finite index in ∆ + = Z p * Z q by using a connection between free subgroups of ∆ + and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy.We also study the generating series for conjugacy classes of free subgroups of finite index in ∆ + , which correspond to isomorphism classes of hypermaps. Asymptotic formulas are provided for the numbers of free subgroups of given finite index, conjugacy classes of such subgroups, or, equivalently, various types of hypermaps and their isomorphism classes.

show abstract

Statistics of isomorphism types in free products

Cited by 9 publications

References 11 publications

Statistics of subgroups of the modular group

Statistics of subgroups of the modular group

Three-dimensional maps and subgroup growth

Free subgroups of free products and combinatorial hypermaps

Contact Info

Product

Resources

About